Beginning of Section
Random1
Theorem.
(
conj_Random1_TMQEe9FjmdtA77CJm8oxVjytEWYiCJos42z
)
∀X0
∈
∅
,
∀X1
⊆
V_
∅
,
(
(
∃X2 ∈
X1
,
∀X3
∈
X0
,
(
∃X4 ∈
X3
,
ordinal
X3
)
→
(
(
∀X4
⊆
X3
,
atleast4
X2
)
∧
(
∃X4 :
set
,
(
(
SNo
X1
→
(
(
(
exactly4
X4
→
(
¬
exactly3
X3
)
)
→
atleast4
X4
)
∧
(
(
(
¬
equip
X1
X1
)
→
nat_p
X3
)
→
(
¬
atleast5
∅
)
)
)
)
∧
atleast2
X3
)
)
)
→
(
¬
setsum_p
X2
)
)
∧
(
∃X2 :
set
,
(
TransSet
X1
∧
(
∀X3
∈
X2
,
setsum_p
X3
→
(
∀X4
⊆
X2
,
(
atleast6
X4
∧
(
(
(
(
¬
ordinal
X3
)
→
atleast6
X3
→
(
¬
exactly3
X2
)
)
∧
exactly3
X2
)
∧
(
¬
atleast4
X0
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
06991f...
and proposition id is
19a433...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMdoKBqRnr4o7BYZoP6j6d4JPMkydEKKA3F
)
∃X0 :
set
,
(
(
∀X1 :
set
,
(
∃X2 :
set
,
(
(
∀X3 :
set
,
(
¬
TransSet
∅
)
)
∧
(
∀X3 :
set
,
(
¬
exactly4
X0
)
→
(
∀X4
⊆
X2
,
ordinal
X3
→
atleast5
X3
)
)
)
)
→
(
∃X2 :
set
,
∃X3 :
set
,
(
¬
TransSet
X0
)
→
(
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
(
exactly3
X2
∧
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
∀X2 :
set
,
(
∀X3
∈
X1
,
∃X4 :
set
,
(
(
X4
⊆
X0
)
∧
atleast5
X1
)
)
→
(
∃X3 :
set
,
∀X4
∈
X3
,
(
¬
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
exactly3
X6
∧
atleast3
X4
)
)
)
→
(
(
(
¬
nat_p
X1
)
∧
(
(
¬
TransSet
X4
)
→
(
¬
atleast3
X3
)
)
)
∧
(
(
¬
nat_p
X4
)
∧
atleast6
X3
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
067361...
and proposition id is
7d335c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMH4ZacEkoR1PKiEftwQYVDcNKBwmTu9Msn
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
∃X1 :
set
,
(
(
∀X2
⊆
X1
,
∀X3 :
set
,
(
∃X4 :
set
,
(
(
X4
⊆
setsum
∅
X0
)
∧
(
(
nat_p
X2
→
(
¬
nat_p
∅
)
)
∧
(
(
¬
tuple_p
X3
X4
)
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
nat_p
X6
)
)
→
exactly3
X3
)
→
(
¬
exactly3
X3
)
→
(
(
SNoLt
X3
X2
∧
nat_p
X3
)
∧
(
(
atleast6
X3
∧
(
(
nat_p
X3
∧
atleast3
X2
)
∧
(
TransSet
X3
∧
exactly2
X2
)
)
)
∧
exactly5
X3
)
)
)
)
)
)
)
→
(
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
nat_p
X4
)
)
)
→
(
∃X4 :
set
,
(
X2
⊆
X2
)
→
atleast4
X4
→
(
(
(
(
¬
tuple_p
X3
X3
)
→
(
¬
atleast3
∅
)
)
→
(
¬
atleast4
X2
)
)
∧
(
¬
exactly5
X0
)
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
∀X3 :
set
,
(
¬
nat_p
X3
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
710a33...
and proposition id is
9063f8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMNEa7dj5GTMAMRKgDFwhDUijDxdCcgJHbt
)
∀X0 :
set
,
(
∀X1 :
set
,
exactly5
X0
→
(
∃X2 :
set
,
(
(
∀X3
∈
SNoLev
X2
,
(
¬
atleast5
X1
)
→
(
∀X4 :
set
,
(
(
(
(
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
SNoLe
X4
X6
)
→
(
¬
exactly4
(
𝒫
X3
)
)
)
∧
(
(
(
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
atleast3
X3
∧
ordinal
X4
)
)
∧
(
¬
atleast4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
(
(
(
(
(
¬
exactly2
X2
)
→
(
exactly5
(
PSNo
X2
(
λX5 :
set
⇒
(
¬
set_of_pairs
∅
)
→
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
SNo_
X7
∅
)
)
)
)
)
∧
(
¬
atleast4
X3
)
)
)
∧
(
¬
atleast6
X3
)
)
∧
(
exactly5
X4
∧
(
TransSet
X2
∧
(
¬
exactly2
X0
)
)
)
)
∧
(
(
(
¬
atleast4
(
binintersect
X3
X4
)
)
→
PNo_upc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
atleast6
X3
∧
X6
X5
)
)
X3
(
λX5 :
set
⇒
(
¬
exactly4
X3
)
)
)
→
nat_p
X4
)
)
∧
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
exactly3
∅
∧
(
¬
setsum_p
X4
)
)
)
)
→
(
(
¬
set_of_pairs
X4
)
∧
(
¬
atleast5
X4
)
)
)
∧
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
∀X3
∈
binrep
X2
X0
,
atleast4
X2
→
(
¬
exactly4
X2
)
)
)
)
)
→
(
∀X1 :
set
,
exactly2
X1
→
(
∀X2 :
set
,
(
∀X3
⊆
X2
,
(
(
∃X4 ∈
Sing
X3
,
(
¬
atleast2
X1
)
)
∧
atleast6
X3
)
→
(
∀X4
∈
ordsucc
X1
,
(
¬
atleast5
X4
)
)
)
→
(
∃X3 :
set
,
exactly2
X2
→
(
¬
SNoEq_
X1
X1
X2
)
)
)
)
In Proofgold the corresponding term root is
44fdb9...
and proposition id is
30273e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMXTV4Fw4XHL9DUv41yYzCoLRAefmmx8ohs
)
∃X0 :
set
,
∃X1 :
set
,
(
(
X1
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
exactly4
X0
→
(
∃X2 ∈
X0
,
exactly3
X1
→
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
nat_p
X3
∧
(
(
¬
(
X3
∈
X4
)
)
→
(
¬
atleast6
X4
)
)
)
)
∧
(
(
∃X4 ∈
X1
,
atleast5
X4
)
∧
(
(
(
(
∃X4 :
set
,
(
inj
X2
X4
(
λX5 :
set
⇒
X4
)
∧
(
¬
set_of_pairs
X4
)
)
)
∧
(
¬
ordinal
∅
)
)
→
(
¬
setsum_p
∅
)
)
∧
(
∀X4
∈
⋃
X2
,
atleast2
X4
)
)
)
)
)
)
→
(
(
∀X2 :
set
,
∀X3 :
set
,
(
¬
equip
X2
X3
)
→
(
∀X4 :
set
,
(
¬
TransSet
X3
)
)
)
∧
(
∃X2 :
set
,
∃X3 :
set
,
∀X4
∈
mul_nat
X2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
,
(
¬
atleast6
X3
)
)
)
)
)
In Proofgold the corresponding term root is
5528c2...
and proposition id is
882405...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMj5ARG9BXR3XTjwFRDiUnbZ57NDYvUCGL
)
∀X0 :
set
,
∀X1
⊆
X0
,
∀X2
⊆
X1
,
∀X3
∈
X0
,
(
∀X4 :
set
,
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
exactly5
X3
)
)
→
(
∃X4 :
set
,
(
(
(
(
(
¬
SNo
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
set_of_pairs
(
V_
X0
)
)
)
)
∧
(
(
(
(
(
(
¬
atleast5
X3
)
→
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast6
X4
→
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
(
(
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
TransSet
X0
)
→
(
(
(
(
(
(
atleast4
∅
∧
(
(
(
(
(
(
(
¬
exactly3
X6
)
∧
(
(
(
¬
exactly5
X5
)
→
(
(
¬
nat_p
X5
)
∧
nat_p
X3
)
→
(
¬
atleast6
X5
)
)
→
(
(
(
(
¬
nat_p
X2
)
∧
(
(
(
exactly3
X6
∧
(
atleast6
X5
→
(
ordinal
X0
→
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
¬
exactly4
(
setsum
X6
X0
)
)
)
)
)
→
atleast6
X6
)
∧
atleast2
X6
)
)
∧
TransSet
X5
)
∧
(
atleast4
X0
→
(
Inj0
X5
∈
setexp
X5
X0
)
)
)
)
)
∧
(
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
(
¬
exactly2
X6
)
→
(
(
(
X0
=
X2
)
→
(
¬
(
X0
⊆
X5
)
)
→
(
(
atleast4
X6
→
(
¬
exactly3
X0
)
)
∧
(
(
(
X0
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
(
(
¬
atleast4
(
SNoLev
X6
)
)
→
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
SNo
X4
∧
(
¬
atleast3
X7
)
)
∧
(
¬
TransSet
X2
)
)
→
atleast4
X7
)
)
→
(
SNoLt
X3
X3
∧
(
atleast2
X0
∧
exactly4
X1
)
)
)
∧
(
¬
atleast6
X6
)
)
)
→
(
(
(
(
¬
TransSet
X0
)
∧
(
(
exactly2
(
Inj1
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
atleastp
X5
X0
∧
(
PNoEq_
X5
(
λX7 :
set
⇒
atleast6
X6
)
(
λX7 :
set
⇒
(
(
(
¬
TransSet
∅
)
∧
(
¬
atleast3
X0
)
)
∧
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
atleast3
X6
)
)
→
(
¬
exactly5
X5
)
→
exactly5
X6
→
(
(
¬
exactly5
X6
)
∧
(
¬
exactly2
X0
)
)
)
)
)
→
(
¬
exactly2
X0
)
→
atleast5
X0
)
)
→
(
nat_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
nat_p
X6
)
→
(
¬
atleast3
X4
)
)
∧
(
¬
SNo_
X5
X6
)
)
)
)
)
→
exactly4
X5
)
→
(
setsum_p
X6
∧
(
(
(
¬
exactly2
X5
)
→
(
(
(
(
¬
atleast2
X6
)
∧
(
X5
=
X0
)
)
→
(
¬
atleast6
X3
)
)
∧
(
atleast5
∅
∧
(
SNoLe
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
¬
atleast4
∅
)
∧
(
exactly3
X1
→
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
ordinal
X5
)
)
)
)
)
→
exactly2
X4
)
)
)
∧
(
PNo_upc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
(
¬
X8
X0
)
→
(
(
¬
atleast4
X6
)
→
X8
X7
)
→
(
(
(
(
¬
X8
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
¬
X8
X6
)
∧
atleast4
X2
)
)
∧
X8
X6
)
∧
atleast5
X6
)
→
(
¬
X8
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
X2
(
λX7 :
set
⇒
(
¬
tuple_p
X6
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
(
X3
∈
X7
)
)
∧
(
¬
(
X6
∈
X6
)
)
)
)
)
)
→
(
(
(
exactly5
X6
∧
exactly3
(
SNoElts_
X5
)
)
∧
exactly3
X5
)
∧
(
¬
exactly2
(
nat_primrec
X6
(
λX7 :
set
⇒
λX8 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
X6
)
)
)
)
∧
(
(
(
¬
ordinal
X5
)
→
atleast6
∅
)
→
atleast5
X1
)
)
→
atleast3
X6
)
∧
ordinal
X5
)
)
→
(
¬
SNo_
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X0
)
)
∧
exactly5
X1
)
∧
(
¬
atleast2
X5
)
)
∧
(
¬
exactly2
X6
)
)
∧
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
¬
ordinal
X1
)
→
(
(
(
¬
exactly3
X5
)
→
(
(
(
¬
ordinal
X0
)
∧
exactly5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
atleast6
X4
)
)
)
∧
(
(
¬
ordinal
X5
)
→
(
(
(
(
atleast4
∅
∧
(
(
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
exactly3
X5
→
(
SNo
X5
∧
(
(
(
(
(
¬
nat_p
(
𝒫
X6
)
)
→
(
¬
exactly2
X6
)
)
→
(
¬
atleast6
X0
)
)
∧
(
set_of_pairs
X0
→
(
¬
exactly3
X5
)
)
)
→
(
¬
TransSet
X5
)
)
)
)
∧
(
¬
setsum_p
X2
)
)
)
∧
(
(
¬
atleast2
X6
)
→
(
¬
SNo
(
ap
X6
X6
)
)
)
)
∧
(
¬
atleast6
X2
)
)
∧
(
¬
atleast2
X6
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
exactly5
X4
)
→
(
(
ordinal
X4
→
set_of_pairs
X3
)
∧
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
¬
setsum_p
X1
)
→
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
(
(
(
(
¬
nat_p
X3
)
∧
(
¬
exactly3
X4
)
)
→
(
¬
SNo
X4
)
)
∧
(
(
¬
atleast5
X4
)
→
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
exactly5
X1
)
→
(
¬
inj
∅
∅
(
λX5 :
set
⇒
X0
)
)
)
∧
atleast5
X3
)
)
→
TransSet
X0
)
→
(
∅
∈
X4
)
)
∧
(
¬
atleast5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
¬
(
X3
∈
X4
)
)
)
→
(
(
equip
∅
X4
∧
(
¬
exactly2
X3
)
)
→
(
(
(
(
atleast4
∅
∧
(
(
(
(
(
(
(
¬
exactly1of2
(
TransSet
X2
)
(
(
¬
atleast5
X1
)
→
(
¬
SNo
X0
)
→
(
¬
exactly3
X1
)
→
(
(
¬
exactly4
X4
)
∧
set_of_pairs
X2
)
)
)
∧
exactly5
X2
)
→
(
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
set_of_pairs
X4
)
)
)
∧
(
¬
atleast6
X0
)
)
∧
(
¬
setsum_p
X3
)
)
→
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
atleast3
X4
∧
(
¬
atleast3
X2
)
)
→
ordinal
X0
)
)
)
∧
nat_p
X0
)
∧
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
exactly4
∅
)
→
(
(
¬
atleast4
X3
)
∧
(
¬
exactly3
X3
)
)
)
→
(
¬
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
exactly4
(
famunion
X4
(
λX5 :
set
⇒
X4
)
)
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
(
exactly2
X5
→
(
¬
(
X6
∈
X6
)
)
)
∧
(
(
¬
(
X5
=
X5
)
)
∧
(
¬
tuple_p
X2
X5
)
)
)
→
(
¬
atleast6
X5
)
)
∧
(
atleast4
X4
→
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
)
∧
atleast6
(
V_
X1
)
)
→
nat_p
X3
)
)
→
atleast5
X3
)
→
(
ordinal
X2
→
(
(
(
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
¬
nat_p
(
V_
X3
)
)
)
→
atleast2
X4
→
(
TransSet
∅
→
ordinal
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
atleast3
X1
)
)
∧
(
(
(
X3
∈
X3
)
→
(
¬
exactly3
(
V_
X4
)
)
→
(
(
¬
atleast6
X4
)
∧
(
nat_p
X2
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
∧
(
(
exactly4
X4
∧
(
(
(
(
ordinal
X3
→
(
X4
∈
X1
)
)
→
exactly4
X0
)
→
(
TransSet
X4
∧
(
¬
SNo
X2
)
)
)
→
bij
X4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
(
λX5 :
set
⇒
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
(
¬
atleast3
X4
)
→
(
¬
atleast2
X2
)
)
∧
(
exactly2
X2
→
nat_p
X1
)
)
)
)
)
)
→
atleast3
X4
→
(
exactly3
∅
∧
SNo_
X1
X1
)
→
(
(
¬
atleast6
X3
)
∧
atleast5
X2
)
)
→
(
¬
atleast2
∅
)
→
(
atleast2
(
Inj0
X1
)
→
(
¬
ordinal
X3
)
)
→
(
¬
ordinal
X2
)
)
)
→
exactly3
X1
)
∧
(
(
(
setsum_p
(
𝒫
∅
)
∧
(
¬
TransSet
X3
)
)
→
(
(
¬
(
X2
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
exactly2
X0
∧
(
¬
exactly2
X3
)
)
)
)
→
(
(
exactly4
X4
→
exactly2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
¬
exactly3
X4
)
→
SNo
X3
)
)
)
)
)
In Proofgold the corresponding term root is
450606...
and proposition id is
76af3c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMZjzw8x5Z73f1PcAbVmsQLBdguh5331NWw
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
(
(
¬
atleast5
X0
)
→
(
(
∀X2
⊆
X1
,
∃X3 :
set
,
(
∀X4 :
set
,
atleast2
X2
→
(
¬
PNoLt
X0
(
λX5 :
set
⇒
(
(
atleast2
X3
∧
atleast2
X2
)
∧
(
atleast6
X0
→
SNoEq_
X4
X1
X1
)
)
)
X1
(
λX5 :
set
⇒
ordinal
X0
→
(
¬
atleast3
X5
)
)
)
)
→
(
∃X4 ∈
X2
,
SNo_
(
setminus
X4
X4
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
∀X2 :
set
,
∀X3
⊆
∅
,
∀X4
∈
X0
,
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
∀X2
∈
X1
,
∀X3 :
set
,
(
¬
bij
X3
(
⋃
X1
)
(
λX4 :
set
⇒
X3
)
)
→
(
(
∀X4
∈
binunion
X3
X3
,
(
¬
exactly3
∅
)
→
(
¬
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast5
X6
)
)
)
)
→
(
∃X4 :
set
,
(
¬
exactly2
X4
)
)
)
→
(
atleast4
X1
∧
atleast3
∅
)
→
(
¬
ordinal
X0
)
)
)
)
)
→
(
∀X1
∈
X0
,
(
atleast5
X1
→
(
∀X2 :
set
,
(
∀X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
setsum_p
(
𝒫
X3
)
)
)
)
→
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
∃X2 ∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
,
∃X3 :
set
,
(
(
X3
⊆
∅
)
∧
(
(
exactly3
X2
→
(
∀X4 :
set
,
(
¬
set_of_pairs
X4
)
→
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
∃X4 :
set
,
exactly2
X3
)
)
)
)
)
In Proofgold the corresponding term root is
57229c...
and proposition id is
ac4aaa...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMFFathu6TBSjmpNX4gtNvq7gUX6UzeGuwJ
)
∃X0 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
,
∀X1 :
set
,
(
¬
bij
∅
X1
(
λX2 :
set
⇒
X0
)
)
→
(
∃X2 ∈
X1
,
(
(
∀X3 :
set
,
(
(
∀X4 :
set
,
(
¬
TransSet
X4
)
)
→
(
∃X4 :
set
,
(
(
ordinal
X4
→
(
¬
exactly3
X2
)
)
∧
atleast2
X4
)
)
)
→
(
∃X4 :
set
,
(
¬
atleast4
X3
)
)
→
(
∃X4 :
set
,
(
(
¬
exactly3
X4
)
∧
(
(
¬
TransSet
X2
)
∧
(
(
(
(
(
(
¬
exactly2
∅
)
→
(
(
atleastp
X4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
SNo_
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X1
)
∧
(
(
(
(
(
TransSet
(
SetAdjoin
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X2
)
∧
(
(
¬
atleast5
X4
)
→
(
¬
atleast6
X3
)
)
)
∧
SNoLe
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
X1
)
→
(
¬
atleast6
∅
)
)
∧
(
(
ordinal
X0
∧
(
(
¬
atleast4
X4
)
→
(
(
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
nat_p
∅
)
)
→
(
¬
ordinal
X3
)
)
→
(
(
(
(
(
(
(
atleast2
X3
∧
(
¬
ordinal
X4
)
)
∧
(
(
(
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
TransSet
X2
)
)
∧
TransSet
X2
)
∧
(
(
¬
setsum_p
X0
)
→
(
¬
exactly5
X0
)
)
)
)
→
SNo_
X4
X4
)
→
(
¬
PNoEq_
X2
(
λX5 :
set
⇒
exactly2
X4
)
(
λX5 :
set
⇒
atleast2
∅
)
)
)
∧
atleast5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
(
(
¬
atleast2
X3
)
∧
(
¬
exactly2
X3
)
)
∧
(
(
(
¬
exactly4
X4
)
→
(
¬
exactly3
X4
)
)
∧
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
(
¬
transitive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
set_of_pairs
∅
)
)
∧
(
¬
SNo
X3
)
)
)
)
→
(
(
¬
atleast3
X0
)
∧
(
(
¬
ordinal
X3
)
→
atleast6
X4
)
)
)
)
)
∧
(
X3
∈
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
atleast4
X3
)
)
→
(
¬
exactly2
X3
)
)
→
TransSet
X4
)
)
)
)
)
∧
(
(
(
∀X3 :
set
,
(
(
∀X4 :
set
,
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
∀X4
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
exactly3
(
proj0
(
ordsucc
X4
)
)
)
)
)
∧
(
(
(
∀X3 :
set
,
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
ordinal
X0
)
)
)
→
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
∀X3 :
set
,
(
∃X4 ∈
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
(
(
(
¬
atleast2
X4
)
∧
(
(
atleast5
X0
∧
(
¬
setsum_p
X3
)
)
→
(
¬
set_of_pairs
X3
)
)
)
∧
(
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly5
X3
)
→
(
¬
atleast3
X3
)
)
)
)
→
(
∀X4 :
set
,
(
(
(
(
¬
atleast6
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast4
X3
→
(
¬
exactly2
∅
)
→
nat_p
X4
)
→
(
¬
(
X3
⊆
X2
)
)
)
∧
exactly5
X3
)
)
)
∧
(
∃X3 ∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
∃X4 :
set
,
(
(
¬
atleast3
(
ordsucc
X4
)
)
∧
(
(
¬
atleast6
X3
)
∧
(
atleast4
X3
→
(
¬
nat_p
X4
)
)
)
)
→
(
(
¬
SNo
X3
)
→
(
(
(
¬
set_of_pairs
X2
)
∧
(
(
ordinal
X1
∧
atleast6
X2
)
∧
(
(
¬
SNoLe
X4
X1
)
→
(
¬
exactly3
X3
)
)
)
)
∧
(
X2
∈
∅
)
)
)
→
exactly2
X4
)
)
∧
(
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
atleast5
X3
)
)
∧
(
(
∀X3 :
set
,
exactly5
(
setexp
X3
X0
)
→
(
∀X4
⊆
X1
,
atleast3
∅
)
)
∧
atleast4
(
binunion
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
∧
(
∀X3 :
set
,
(
∃X4 ∈
X3
,
(
¬
atleast6
X2
)
)
→
(
(
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
atleast6
X3
→
(
¬
(
X1
=
X2
)
)
)
∧
(
¬
exactly3
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
f3128e...
and proposition id is
fa5437...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMQMg2kXf9ajAA64RMWejZiByGaq16JZjec
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
∃X1 :
set
,
∀X2 :
set
,
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
∀X3
⊆
X1
,
∃X4 ∈
X1
,
(
(
¬
TransSet
X3
)
→
(
¬
ordinal
∅
)
)
→
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly5
X5
)
)
)
)
In Proofgold the corresponding term root is
07140d...
and proposition id is
2c5723...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMT6GyM8a3pd6cjcxix5N31bct2iVwg6AGo
)
∀X0 :
set
,
∃X1 ∈
X0
,
∀X2 :
set
,
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
∃X3 ∈
X2
,
∀X4
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
exactly5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
¬
SNo_
X4
X4
)
)
∧
(
(
¬
atleast6
(
⋃
X4
)
)
→
(
¬
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast2
X4
)
→
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
nat_p
X0
)
)
)
)
)
In Proofgold the corresponding term root is
cd4e01...
and proposition id is
c9a303...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMc9p14P2JPeE9w1Ut7EY4nunuv6QvgYyiq
)
∀X0 :
set
,
∀X1 :
set
,
(
(
(
∃X2 ∈
X1
,
∀X3 :
set
,
(
∀X4 :
set
,
atleast4
X1
)
→
(
∀X4 :
set
,
exactly4
X3
)
)
→
(
∃X2 :
set
,
(
(
(
(
(
∃X3 ∈
X2
,
∃X4 :
set
,
atleast3
∅
)
→
(
¬
nat_p
X2
)
)
→
(
X1
∈
∅
)
)
→
(
∀X3
∈
X2
,
PNoEq_
X2
(
λX4 :
set
⇒
(
¬
atleast2
X0
)
→
(
¬
atleast4
X4
)
)
(
λX4 :
set
⇒
(
¬
atleast6
X1
)
)
)
)
∧
set_of_pairs
X2
)
)
)
∧
ordinal
X1
)
In Proofgold the corresponding term root is
40e6b9...
and proposition id is
beafe5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMzQ9L9zvDmcxKHCRUNmc5E4fikdNpQabt
)
∃X0 :
set
,
(
(
X0
⊆
setsum
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
∀X1
⊆
X0
,
∀X2
∈
X1
,
∀X3 :
set
,
(
∃X4 :
set
,
(
(
¬
ordinal
X0
)
∧
(
atleast3
X0
→
(
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
nat_p
X3
)
∧
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
∃X4 ∈
X2
,
(
¬
exactly5
X4
)
)
)
)
In Proofgold the corresponding term root is
d460de...
and proposition id is
a138d6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMQ6BwNPr6YbEkiJYpkGnYcrbBM8ZiqS2b
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
∃X1 ∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
,
(
(
atleast6
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
∀X2
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
,
∀X3 :
set
,
(
∃X4 :
set
,
(
exactly2
(
binunion
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
X1
)
∧
(
¬
atleast5
(
Pi
X2
(
λX5 :
set
⇒
X5
)
)
)
)
)
→
(
∀X4 :
set
,
(
¬
ordinal
X2
)
)
)
)
∧
(
(
∃X2 ∈
X1
,
∃X3 :
set
,
(
(
∃X4 :
set
,
(
(
(
(
¬
exactly3
X3
)
→
(
(
(
¬
setsum_p
X4
)
∧
set_of_pairs
X3
)
→
(
X3
∈
X3
)
)
→
(
exactly3
∅
∧
exactly5
X2
)
)
→
atleast2
X2
)
∧
atleast5
X4
)
)
∧
(
¬
exactly5
X3
)
)
)
→
(
¬
setsum_p
X0
)
)
)
)
)
In Proofgold the corresponding term root is
42ec0d...
and proposition id is
3ee7e5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSMwB2rQK8zU3VYWzTHVTwfVkQDB6KRUPH
)
∀X0 :
set
,
∀X1
∈
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
∀X2 :
set
,
exactly1of3
(
(
¬
exactly2
X0
)
→
(
¬
TransSet
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
(
(
∀X3 :
set
,
(
∅
∈
X3
)
→
(
∃X4 ∈
X0
,
atleastp
X1
(
ordsucc
X3
)
)
)
∧
(
∀X3 :
set
,
(
(
∃X4 ∈
X0
,
(
(
¬
ordinal
(
⋃
X4
)
)
∧
equip
∅
X3
)
→
(
atleast3
X3
∧
atleast5
X0
)
→
(
exactly5
X4
∧
(
(
¬
atleast4
X3
)
∧
(
(
(
(
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
(
¬
SNo
X4
)
)
∧
(
atleast2
X2
→
(
(
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
=
X0
)
)
)
→
SNo
X3
)
∧
(
¬
exactly2
X3
)
)
→
exactly2
X1
)
∧
(
(
(
(
¬
atleast3
X4
)
→
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
exactly2
X4
∧
(
atleast3
X2
→
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
(
¬
exactly2
X3
)
∧
exactly4
X4
)
)
)
)
)
∧
(
(
¬
ordinal
X2
)
∧
(
¬
atleast3
X3
)
)
)
∧
(
¬
atleast4
X2
)
)
)
)
)
→
setsum_p
X2
)
→
(
(
(
∀X4
⊆
X0
,
(
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast4
X4
)
∧
exactly2
(
𝒫
X2
)
)
)
∧
(
∃X4 ∈
X3
,
exactly4
X3
)
)
∧
(
∀X4
⊆
X0
,
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
(
∃X3 :
set
,
(
(
¬
(
X0
∈
X2
)
)
∧
(
(
(
(
X3
∈
∅
)
∧
set_of_pairs
X2
)
→
(
∃X4 ∈
proj1
X3
,
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
¬
atleast6
∅
)
)
→
(
∀X4
∈
∅
,
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
9f74db...
and proposition id is
66f31b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMWLU34YkyCYKsjFr2a6ViPZr1Fnb7vyNid
)
∀X0
⊆
⋃
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
,
∃X1 :
set
,
(
(
X1
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
atleast5
X1
→
(
¬
reflexive_i
(
λX2 :
set
⇒
λX3 :
set
⇒
(
∀X4 :
set
,
(
(
SNoElts_
X3
=
∅
)
∧
(
¬
exactly2
X4
)
)
)
→
(
∃X4 ∈
X0
,
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
60dd33...
and proposition id is
b2a29c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMTw4ZySVFz6L3ptJFjfApKNczeWZVVnPX6
)
∀X0
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
∀X1 :
set
,
(
(
∀X2
∈
X1
,
(
¬
atleast5
X1
)
)
→
atleast4
X1
)
→
(
∀X2
∈
X0
,
∃X3 :
set
,
(
(
X3
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
atleast6
X3
∧
equip
X3
X1
)
)
)
)
)
In Proofgold the corresponding term root is
cd8b6e...
and proposition id is
009b6c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMRt5v89TYM3NEb9rg2LAUDmKB47XoRMovU
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
X1
⊆
∅
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
reflexive_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
¬
atleast5
∅
)
)
)
)
)
)
→
(
∀X1 :
set
,
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3 :
set
,
(
∃X4 ∈
X1
,
exactly5
X2
→
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
∃X4 :
set
,
(
(
∅
∈
X4
)
∧
(
(
(
(
¬
SNoEq_
X4
X4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
(
¬
set_of_pairs
X3
)
→
(
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
¬
atleast5
X1
)
)
→
nat_p
X4
)
)
∧
(
(
¬
SNo
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
TransSet
(
Repl
X3
(
λX5 :
set
⇒
X5
)
)
→
binop_on
X2
(
λX5 :
set
⇒
λX6 :
set
⇒
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
exactly3
X4
)
)
)
∧
(
¬
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast3
∅
)
)
)
)
)
)
)
)
→
(
∃X2 :
set
,
(
(
(
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
∀X3
⊆
X2
,
(
∃X4 :
set
,
(
(
¬
TransSet
X3
)
∧
(
¬
set_of_pairs
X4
)
)
)
→
(
∀X4
⊆
X2
,
(
(
¬
atleast2
X4
)
∧
exactly5
X4
)
)
)
→
(
∃X3 :
set
,
(
∀X4 :
set
,
SNoLt
X3
X4
→
(
X1
⊆
X2
)
)
→
(
∀X4
∈
X2
,
exactly5
X4
→
atleast5
X2
)
→
exactly2
X3
→
(
∃X4 ∈
X3
,
(
(
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
nat_p
X4
→
exactly3
X0
)
)
∧
(
(
¬
set_of_pairs
X3
)
∧
(
¬
atleast3
(
lam2
(
Sing
X4
)
(
λX5 :
set
⇒
X4
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X5
)
)
)
)
)
)
)
)
)
∧
(
(
∀X3
∈
proj1
X1
,
(
(
∃X4 ∈
X3
,
(
¬
exactly5
X4
)
)
∧
(
∀X4 :
set
,
set_of_pairs
X4
)
)
)
∧
(
(
∀X3
⊆
X1
,
∃X4 :
set
,
(
(
X4
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
atleast3
(
ordsucc
X1
)
)
)
→
(
∃X3 ∈
X0
,
∃X4 :
set
,
(
exactly4
X1
∧
(
(
(
¬
ordinal
X4
)
→
(
X3
∈
X3
)
→
(
¬
exactly5
X3
)
)
∧
setsum_p
(
⋃
∅
)
)
)
)
)
)
)
∧
(
(
∃X3 ∈
∅
,
(
∃X4 :
set
,
(
(
¬
exactly2
X4
)
∧
(
¬
TransSet
X3
)
)
)
→
(
∃X4 ∈
X3
,
(
¬
atleast5
X3
)
)
)
→
(
∃X3 ∈
𝒫
(
Inj1
X1
)
,
∃X4 ∈
X2
,
(
¬
atleast4
(
setsum
X2
X4
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3ea84c...
and proposition id is
6d2aac...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMQQ31YUpcVhV3rkqnbTiPz9W1VZYee6M3F
)
∃X0 :
set
,
∃X1 :
set
,
(
(
X1
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∧
(
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
X3
⊆
Inj0
X2
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
(
¬
atleast2
X4
)
)
)
)
)
∧
(
∃X3 :
set
,
(
(
¬
nat_p
X2
)
∧
(
∃X4 :
set
,
(
(
X3
⊆
X4
)
∧
(
¬
setsum_p
X1
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
478ddd...
and proposition id is
098236...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMP2f75JQo596LCr2DLo34nSJ3zCmYi8otw
)
∀X0
∈
∅
,
∃X1 :
set
,
(
nat_p
X1
∧
(
∃X2 :
set
,
(
(
∃X3 ∈
X0
,
(
(
¬
set_of_pairs
X1
)
∧
(
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
SNoLe
X3
X4
→
(
¬
SNoEq_
X2
X2
X3
)
)
→
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
¬
atleast6
X5
)
→
(
(
(
¬
atleast6
X5
)
→
(
(
(
(
(
(
¬
exactly5
X6
)
→
(
X5
∈
If_i
(
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
equip
X1
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
atleastp
X7
X2
→
(
(
(
¬
atleast3
X7
)
→
(
¬
ordinal
X7
)
→
(
(
(
¬
atleast3
X7
)
∧
(
¬
ordinal
X7
)
)
∧
(
(
¬
exactly4
X3
)
∧
exactly3
X6
)
)
)
∧
(
X7
∈
X7
)
)
)
)
→
(
(
¬
atleast5
X6
)
∧
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
X5
X6
)
)
∧
(
¬
exactly5
X6
)
)
∧
(
¬
TransSet
X5
)
)
∧
(
¬
atleast2
X5
)
)
→
(
(
(
(
(
¬
exactly2
X0
)
∧
atleast6
X6
)
∧
ordinal
X6
)
→
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
atleast2
X0
)
)
)
→
(
¬
atleast2
∅
)
)
→
atleast4
X6
)
→
(
¬
per_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
exactly2
X8
→
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
atleast3
X1
∧
(
(
(
¬
SNoLe
X7
X8
)
→
(
X7
∈
X2
)
)
→
exactly5
X8
)
)
→
(
(
X8
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
atleast3
(
ordsucc
X7
)
)
)
)
)
)
)
)
→
(
¬
TransSet
(
ordsucc
X5
)
)
)
)
)
)
→
(
∃X4 ∈
X3
,
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
set_of_pairs
X4
)
∧
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
)
)
)
∧
(
(
(
(
(
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
(
¬
atleast5
X4
)
∧
(
¬
exactly5
X3
)
)
)
∧
(
(
∀X4
⊆
X3
,
(
¬
exactly3
X2
)
)
→
(
∀X4 :
set
,
atleast3
X4
)
)
)
)
∧
(
∀X3
∈
∅
,
∀X4
⊆
X2
,
(
¬
atleast6
X4
)
)
)
→
ordinal
X0
)
∧
(
∀X3
⊆
X1
,
∀X4
⊆
X1
,
(
(
(
¬
exactly3
X3
)
∧
(
exactly4
X2
→
(
(
¬
SNo_
X1
X3
)
∧
(
exactly2
X4
∧
(
exactly3
X0
→
(
¬
exactly3
X0
)
)
)
)
)
)
∧
(
(
(
(
ordinal
X3
∧
(
(
atleast5
X3
→
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
(
(
(
(
¬
exactly2
X1
)
→
(
(
(
¬
atleast4
X0
)
∧
nat_p
X0
)
∧
(
(
X2
∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
atleast6
X4
)
)
)
∧
(
(
(
¬
atleast6
X0
)
∧
(
¬
exactly5
∅
)
)
→
atleast4
X4
→
(
(
¬
atleast4
X3
)
∧
(
(
(
exactly2
X3
→
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
exactly5
∅
→
(
¬
TransSet
X3
)
)
)
)
)
)
→
atleast2
X4
)
∧
(
¬
exactly4
X3
)
)
)
)
→
(
(
(
(
(
nat_p
X2
→
(
(
(
(
¬
TransSet
X4
)
∧
nat_p
X3
)
∧
(
¬
SNo
X0
)
)
∧
atleast4
X3
)
)
∧
(
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
exactly2
X4
)
)
∧
atleast2
X4
)
∧
(
(
(
ordinal
X2
∧
(
(
(
¬
ordinal
X2
)
→
(
¬
nat_p
X3
)
)
→
(
(
¬
exactly2
X3
)
∧
(
¬
atleast5
X4
)
)
)
)
∧
(
(
(
(
exactly2
X2
∧
exactly2
X4
)
∧
(
¬
atleast5
X3
)
)
→
(
(
(
¬
exactly2
X4
)
→
(
(
¬
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast3
X6
)
)
)
∧
(
(
¬
atleast4
X4
)
∧
nat_p
X4
)
)
)
∧
(
¬
SNoLe
∅
X3
)
)
)
∧
(
(
(
¬
SNo
X2
)
→
(
(
¬
exactly5
X0
)
∧
(
atleast5
X4
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
X2
)
)
)
)
∧
(
(
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
(
(
nat_p
X4
∧
(
(
(
(
exactly2
X4
→
(
¬
exactly3
X2
)
)
→
(
¬
atleast5
(
Sing
X3
)
)
)
→
(
(
¬
atleast5
X4
)
∧
atleast4
(
setminus
X3
X3
)
)
→
(
(
(
¬
ordinal
X3
)
∧
(
¬
exactly4
X4
)
)
∧
nat_p
X4
)
)
→
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
TransSet
X6
)
)
)
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
tuple_p
X6
X5
→
exactly3
X0
→
(
(
(
(
(
(
¬
SNo
X5
)
→
(
¬
atleast6
X6
)
)
∧
(
¬
irreflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
SNoEq_
X7
(
⋃
X7
)
X7
)
)
)
)
∧
(
¬
ordinal
(
binunion
X3
∅
)
)
)
→
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast6
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly4
X7
)
)
)
∧
(
¬
atleast6
X3
)
)
→
(
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
ordinal
X2
)
∧
(
¬
atleast4
X1
)
)
)
∧
(
¬
PNoEq_
X0
(
λX7 :
set
⇒
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
(
atleast6
X8
∧
(
exactly2
X8
→
exactly3
X8
)
)
∧
ordinal
X2
)
)
)
→
exactly3
X5
→
(
¬
atleast2
X1
)
→
(
ordinal
X0
∧
(
(
(
(
¬
equip
X0
∅
)
∧
(
¬
strictpartialorder_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
SNo
X9
∧
(
exactly3
X4
→
atleast4
X8
)
)
)
)
)
∧
exactly5
X2
)
∧
(
¬
atleast5
X0
)
)
)
)
(
λX7 :
set
⇒
atleast2
X1
→
atleast2
X7
)
)
)
)
→
(
(
(
(
exactly3
(
proj0
X0
)
→
ordinal
X6
)
∧
(
atleast3
X0
→
(
(
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
exactly2
X5
)
→
(
¬
nat_p
X0
)
)
→
(
¬
setsum_p
X5
)
)
)
→
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
exactly4
X5
)
)
)
)
)
)
)
)
→
atleast4
X3
)
)
∧
exactly2
X4
)
)
→
(
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
setsum_p
X0
∧
(
¬
nat_p
X4
)
)
→
(
¬
(
X0
∈
X5
)
)
)
→
(
¬
SNo
X6
)
)
→
eqreln_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X4
)
∧
(
exactly5
X8
→
exactly4
X7
)
)
)
)
∧
(
¬
atleast5
X3
)
)
)
→
(
(
¬
exactly5
X4
)
∧
(
equip
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
(
⋃
X4
)
∧
(
atleast5
X2
→
(
¬
exactly5
X4
)
)
)
)
)
)
→
(
¬
atleast5
X4
)
→
(
¬
nat_p
X4
)
)
)
∧
(
∀X3 :
set
,
(
∀X4 :
set
,
(
(
(
(
exactly2
X3
→
(
exactly2
(
V_
X1
)
∧
(
(
TransSet
X1
→
atleast4
X1
)
∧
(
(
(
exactly2
X3
∧
(
(
ordinal
X4
∧
(
(
¬
atleast2
X2
)
→
(
¬
atleast2
X1
)
)
)
∧
(
¬
atleast3
X3
)
)
)
∧
(
(
(
exactly2
X4
→
(
ordinal
X4
∧
(
(
(
¬
exactly4
X0
)
∧
(
¬
atleast3
X3
)
)
∧
(
(
(
(
¬
exactly2
∅
)
∧
(
¬
exactly2
X3
)
)
→
exactly2
X4
→
(
¬
exactly4
X3
)
→
exactly4
∅
→
(
¬
atleast3
X4
)
→
(
¬
atleast3
X4
)
→
(
¬
ordinal
∅
)
)
∧
(
(
¬
atleast5
X2
)
→
TransSet
X2
)
)
)
)
)
∧
atleast6
X4
)
→
(
(
(
¬
nat_p
∅
)
→
(
exactly3
X2
∧
ordinal
X3
)
)
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
ordinal
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∧
ordinal
X5
)
)
∧
atleast3
X4
)
∧
atleast5
X3
)
)
)
)
∧
(
atleast6
(
binrep
X3
X2
)
→
(
(
(
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
exactly4
X1
)
)
∧
atleast3
X3
)
∧
(
¬
exactly3
X3
)
)
)
)
)
)
→
(
¬
atleast2
X2
)
)
→
(
(
¬
atleast2
X3
)
∧
(
¬
exactly4
X4
)
)
)
→
atleast6
X4
)
∧
(
(
(
¬
setsum_p
X2
)
→
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
atleast4
X1
)
)
→
(
atleast2
X0
∧
(
(
(
(
(
¬
atleast6
X4
)
∧
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
TransSet
X3
)
)
∧
(
¬
atleast3
X4
)
)
∧
(
¬
exactly2
∅
)
)
)
)
→
(
¬
exactly3
X2
)
)
)
→
(
¬
(
X1
∈
X0
)
)
→
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
921ef4...
and proposition id is
7cf6bb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSwwEs7ccX6EAeDmZfJEY3GouxRR4mvQnm
)
∃X0 :
set
,
(
(
X0
⊆
ordsucc
(
binunion
∅
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
∀X2 :
set
,
(
(
∃X3 ∈
X0
,
(
¬
exactly2
X3
)
)
→
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
∀X3 :
set
,
(
¬
nat_p
(
binintersect
X2
X3
)
)
→
(
∃X4 :
set
,
(
(
atleast6
(
𝒫
X4
)
→
(
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
(
(
(
atleast5
X3
∧
(
(
¬
nat_p
X4
)
→
(
(
¬
exactly5
(
Inj1
X4
)
)
∧
(
¬
TransSet
X3
)
)
)
)
∧
SNoLe
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
X3
)
→
exactly3
X4
)
)
)
∧
(
¬
ordinal
X3
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
15c78a...
and proposition id is
7a7027...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSbXQjWXSWfvzwkBbDuYj13vY1VfSfCwT2
)
∀X0 :
set
,
(
∀X1
⊆
X0
,
(
exactly2
X1
∧
atleast5
X1
)
)
→
(
∃X1 :
set
,
(
(
X1
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
∃X2 :
set
,
(
(
(
∀X3 :
set
,
(
¬
atleast5
X1
)
)
∧
(
∀X3
⊆
X1
,
∀X4
⊆
X3
,
(
exactly3
X4
∧
(
¬
atleast3
X4
)
)
)
)
∧
(
(
∀X3
⊆
X2
,
(
¬
atleast6
∅
)
→
(
(
∀X4 :
set
,
(
ordinal
X3
→
exactly4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
exactly3
X2
)
∧
(
∃X4 ∈
X3
,
(
¬
atleast5
X1
)
)
)
)
→
(
∀X3
⊆
Inj0
(
SNoLev
X1
)
,
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
(
¬
atleast4
X0
)
→
(
(
(
(
(
¬
(
X4
=
X3
)
)
→
exactly4
X2
)
∧
(
¬
exactly2
X3
)
)
→
TransSet
X4
)
∧
(
(
X2
∈
X0
)
→
SNo_
X3
∅
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
625d7d...
and proposition id is
9a8256...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMHW7SUTU3kEdFw1wAngcdJX2nhVN47oy3N
)
∃X0 :
set
,
∃X1 :
set
,
(
(
X1
⊆
⋃
X0
)
∧
(
∀X2 :
set
,
∀X3
∈
X1
,
(
(
∃X4 ∈
X0
,
(
¬
atleast5
X3
)
)
∧
(
(
∃X4 :
set
,
(
atleast6
X2
∧
(
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
(
¬
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∈
V_
X4
)
)
→
exactly2
(
If_i
(
¬
atleast5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∅
X2
)
)
)
)
)
→
(
∃X4 ∈
X0
,
(
¬
atleast6
X3
)
)
)
)
)
)
In Proofgold the corresponding term root is
c19930...
and proposition id is
3eba1b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMNv5cR4Xcoa372nHbRRzPszdMmozbei54M
)
∀X0 :
set
,
(
∀X1 :
set
,
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
∀X4
∈
X1
,
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
¬
SNo
X2
)
)
)
)
→
(
(
¬
atleast5
X2
)
∧
(
¬
ordinal
X0
)
)
)
)
)
→
(
∀X2 :
set
,
(
(
∀X3
⊆
X2
,
atleast5
X1
)
∧
(
(
(
¬
atleast5
X2
)
∧
(
∀X3
⊆
X2
,
(
(
∃X4 ∈
X2
,
(
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
(
¬
atleast3
X4
)
)
)
∧
(
∀X4 :
set
,
(
(
(
(
atleast5
X2
∧
(
(
¬
atleast3
X2
)
∧
(
(
¬
exactly4
X1
)
→
(
¬
PNoEq_
X2
(
λX5 :
set
⇒
(
nat_p
X5
∧
(
¬
exactly5
∅
)
)
)
(
λX5 :
set
⇒
(
¬
exactly5
X5
)
)
)
→
(
(
¬
atleast2
X2
)
→
(
¬
ordinal
X2
)
)
→
(
(
(
¬
exactly5
X2
)
→
(
(
(
(
TransSet
X0
→
(
¬
exactly2
X0
)
)
∧
(
(
¬
(
X3
∈
mul_nat
X4
∅
)
)
∧
(
(
atleast2
X2
→
(
(
nat_p
X4
∧
(
X4
⊆
X3
)
)
→
nat_p
X3
)
→
(
(
exactly2
X1
→
atleastp
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
X3
)
∧
(
¬
nat_p
X3
)
)
)
→
(
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
exactly4
X4
)
→
(
¬
exactly5
∅
)
)
→
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
nat_p
X3
→
ordinal
X2
)
)
)
∧
(
¬
exactly4
X4
)
)
→
TransSet
X4
→
PNoEq_
X4
(
λX5 :
set
⇒
(
(
atleast3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleastp
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
X2
)
)
∧
(
¬
TransSet
X5
)
)
)
(
λX5 :
set
⇒
(
¬
atleast2
∅
)
)
)
→
setsum_p
X4
)
∧
(
¬
(
X1
⊆
X3
)
)
)
)
)
)
∧
(
¬
exactly4
X4
)
)
∧
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
atleast4
X3
)
→
(
(
(
(
¬
atleast3
(
ordsucc
X3
)
)
→
atleast6
X4
)
→
(
¬
atleast6
X4
)
)
∧
(
¬
nat_p
X3
)
)
)
)
)
)
→
(
∀X3
∈
X2
,
∃X4 ∈
X0
,
(
¬
exactly2
X1
)
)
)
)
→
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
∃X1 :
set
,
∀X2
∈
X1
,
SNo
X1
→
(
∃X3 ∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
,
(
¬
atleast6
X2
)
)
→
exactly5
X1
)
In Proofgold the corresponding term root is
9971ab...
and proposition id is
50e8d2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMXbEUsHrqbPdoSthmj312T2Uh6f8U1M7Dy
)
∃X0 ∈
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
∃X1 :
set
,
(
(
X1
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
atleast3
X1
∧
(
(
∀X2
∈
X1
,
(
∃X3 :
set
,
(
(
X3
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
∃X4 :
set
,
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
∃X3 :
set
,
(
∀X4 :
set
,
(
¬
exactly2
X2
)
→
(
¬
atleast3
X2
)
)
→
(
(
∀X4
⊆
X2
,
atleast6
X3
)
∧
(
∀X4
∈
X3
,
PNoEq_
X3
(
λX5 :
set
⇒
(
¬
exactly4
X3
)
)
(
λX5 :
set
⇒
(
(
(
¬
TransSet
X2
)
→
nat_p
(
⋃
X5
)
)
∧
(
(
¬
exactly5
X4
)
→
(
(
X3
∈
X3
)
∧
(
exactly3
X5
∧
(
(
(
(
atleast6
X5
→
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
setminus
X2
X0
⊆
X5
)
)
∧
(
¬
exactly5
X0
)
)
→
(
¬
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
SNoLe
X4
X2
)
)
)
→
(
¬
atleast5
X1
)
)
)
)
)
)
)
)
→
(
∀X2 :
set
,
(
¬
ordinal
(
⋃
X1
)
)
→
(
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
set_of_pairs
X0
)
)
∧
exactly3
X1
)
→
(
(
∃X3 :
set
,
(
(
X3
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
(
∀X4
⊆
𝒫
X3
,
PNoLt
X4
(
λX5 :
set
⇒
nat_p
X4
)
X1
(
λX5 :
set
⇒
exactly4
X4
)
)
)
)
∧
(
∀X3 :
set
,
exactly5
X0
→
(
X2
∈
X2
)
)
)
)
)
)
∧
(
∀X2 :
set
,
∀X3 :
set
,
atleast5
X3
→
(
¬
atleast5
X3
)
)
)
)
In Proofgold the corresponding term root is
e861fd...
and proposition id is
01dc9a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMFaJhARRpLNaaPtQ4SMpcKYcBZwNaLzfdJ
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
∀X2 :
set
,
∃X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
atleast5
X1
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
set_of_pairs
X1
)
)
)
)
→
(
∀X1
∈
binrep
∅
(
Inj1
∅
)
,
(
¬
reflexive_i
(
λX2 :
set
⇒
λX3 :
set
⇒
exactly4
X0
→
(
(
∀X4
⊆
X2
,
(
¬
atleast4
X2
)
)
∧
(
∀X4
⊆
X1
,
ordinal
X2
)
)
)
)
)
In Proofgold the corresponding term root is
c53b2a...
and proposition id is
79a9ae...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMPbPpvu3p8PKv4e9Wfxn6APE8iFiSuYgqT
)
∃X0 ∈
∅
,
∃X1 :
set
,
(
(
X1
⊆
∅
)
∧
(
TransSet
(
lam2
X0
(
λX2 :
set
⇒
X0
)
(
λX2 :
set
⇒
λX3 :
set
⇒
X2
)
)
∧
(
(
∃X2 ∈
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
(
∃X3 ∈
X0
,
∀X4 :
set
,
(
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly2
X5
)
∧
(
exactly2
X0
∧
(
(
¬
atleast5
X2
)
∧
nat_p
X4
)
)
)
→
(
(
¬
exactly4
X4
)
∧
(
¬
atleast6
X2
)
)
→
(
¬
atleast2
X2
)
→
(
¬
TransSet
X4
)
)
→
(
∀X3 :
set
,
exactly5
X3
)
→
(
∃X3 :
set
,
(
(
(
∃X4 :
set
,
SNo_
X4
X4
)
∧
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
∀X4 :
set
,
atleast6
X2
)
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
∃X3 ∈
X2
,
∃X4 :
set
,
(
(
(
atleast3
X0
→
TransSet
X3
)
∧
(
¬
atleast4
X4
)
)
∧
(
(
¬
TransSet
X3
)
→
(
¬
SNo
X3
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
971fb2...
and proposition id is
6ae4c5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMH1ezGw54iKSSVrWBi1zf5fwURTBZ6KBoJ
)
∀X0
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
,
∀X1 :
set
,
(
∀X2
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
,
(
X1
∈
X2
)
)
→
(
∀X2
⊆
X1
,
∀X3
∈
PSNo
X1
(
λX4 :
set
⇒
∀X5 :
set
,
(
¬
atleast6
X5
)
→
(
¬
atleast6
X4
)
)
,
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
exactly2
X3
∧
(
X1
∈
X2
)
)
∧
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
exactly2
X6
∧
(
¬
atleast6
X5
)
)
∧
(
(
¬
(
X2
∈
X6
)
)
→
atleast6
X0
)
)
)
)
→
(
(
(
atleast4
(
ordsucc
X3
)
∧
(
exactly2
(
⋃
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
atleast5
X4
→
(
X4
∈
X4
)
)
)
)
→
(
¬
ordinal
X3
)
)
∧
(
¬
exactly5
X3
)
)
→
(
ordinal
X0
∧
(
X1
∈
X1
)
)
)
)
)
)
→
(
∀X2 :
set
,
(
∃X3 ∈
proj1
X2
,
∀X4 :
set
,
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast2
X4
)
→
atleast5
X2
)
In Proofgold the corresponding term root is
2bb24b...
and proposition id is
0af13e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVqikVszYaBDyCxvbbXjFeeQfVEwEYKqay
)
∀X0 :
set
,
(
∀X1
⊆
setsum
X0
X0
,
∀X2
⊆
X0
,
∃X3 :
set
,
∃X4 :
set
,
(
(
(
(
ordinal
X4
→
(
¬
setsum_p
X4
)
)
→
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
atleast6
X6
)
∧
(
exactly5
X6
→
(
¬
atleast6
X6
)
)
)
)
)
∧
(
¬
atleast4
X4
)
)
)
∧
(
¬
ordinal
X3
)
)
∧
(
¬
ordinal
X4
)
)
)
→
(
∃X1 :
set
,
(
(
X1
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
∀X2 :
set
,
(
¬
SNoLt
∅
X1
)
)
)
)
In Proofgold the corresponding term root is
e37882...
and proposition id is
5e6e06...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcrkeKGcN5jGcbstFaBW57AQ1Mu5MiF9CF
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
(
∀X2 :
set
,
(
∀X3
⊆
X0
,
atleast4
X2
)
→
(
∃X3 ∈
X2
,
∃X4 :
set
,
TransSet
X2
)
)
∧
(
∃X2 :
set
,
∀X3 :
set
,
atleast3
X2
→
(
∀X4
∈
X2
,
exactly4
X4
→
(
(
(
¬
exactly5
X2
)
∧
(
¬
atleast6
(
Inj0
X0
)
)
)
∧
(
¬
ordinal
(
binunion
X4
X3
)
)
)
)
)
)
)
∧
(
∀X1
∈
X0
,
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3
∈
X2
,
(
¬
atleast2
X2
)
)
)
)
)
In Proofgold the corresponding term root is
d76da3...
and proposition id is
c8c8e3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMXayNheREmDBVA6uHfRp2fc6nQ3PpqVQzy
)
∃X0 :
set
,
∀X1 :
set
,
(
(
∃X2 :
set
,
(
(
¬
atleast5
X1
)
∧
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
(
(
∀X4 :
set
,
exactly5
X4
)
∧
(
∃X4 ∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
(
(
exactly5
X2
∧
TransSet
X4
)
∧
(
(
¬
atleast6
X0
)
∧
(
¬
nat_p
X3
)
)
)
)
)
∧
(
∀X4
⊆
∅
,
(
(
(
(
(
(
atleast4
∅
→
(
(
(
atleast2
X0
∧
(
(
(
(
(
(
exactly3
X1
∧
atleast2
(
add_nat
X2
X3
)
)
∧
atleast5
X3
)
→
atleast3
X3
)
∧
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
∧
(
¬
nat_p
X3
)
)
∧
exactly2
X4
)
)
→
(
¬
TransSet
∅
)
)
∧
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
∧
(
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
tuple_p
X4
X2
)
)
)
∧
(
(
atleast6
X3
→
atleast2
X4
)
→
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
atleast3
X4
)
)
→
(
¬
exactly4
X4
)
)
∧
(
(
X4
∈
X3
)
→
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
¬
atleastp
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
X2
)
)
)
)
)
)
)
)
→
(
¬
ordinal
X0
)
)
→
(
¬
exactly4
X0
)
→
(
∃X2 ∈
X0
,
(
¬
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
In Proofgold the corresponding term root is
22f718...
and proposition id is
4c28fd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSNrz5g75rW7opNZWF8pDb38nziKAZXs3T
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
∀X1 :
set
,
(
(
∃X2 :
set
,
(
equip
X2
X0
∧
(
∃X3 :
set
,
∃X4 :
set
,
(
(
X0
=
X2
)
∧
(
(
¬
atleast6
X2
)
→
(
atleast4
X1
∧
(
¬
exactly2
X2
)
)
)
)
)
)
)
∧
(
¬
set_of_pairs
X0
)
)
→
(
(
¬
symmetric_i
(
λX2 :
set
⇒
λX3 :
set
⇒
∀X4 :
set
,
(
(
(
atleast5
X2
→
(
ordinal
X0
∧
exactly2
X0
)
)
→
(
¬
atleast2
X3
)
)
∧
(
¬
atleast2
(
Inj1
X0
)
)
)
→
(
¬
setsum_p
X4
)
)
)
∧
(
∀X2
∈
X0
,
∀X3
⊆
X1
,
∀X4
⊆
X3
,
TransSet
X3
)
)
)
)
In Proofgold the corresponding term root is
9e5eb8...
and proposition id is
f326a3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSoQEKHfLREbFdTJbTzLVSGdb85qE4D5n9
)
∀X0 :
set
,
(
∀X1 :
set
,
(
∀X2
∈
X0
,
∀X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
binrep
X3
X2
)
∧
(
(
(
atleast5
X0
∧
(
(
¬
setsum_p
X2
)
→
(
¬
exactly5
X0
)
)
)
∧
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
→
(
∀X2 :
set
,
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
∃X4 :
set
,
(
(
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
SNoLe
X6
X6
∧
(
atleast6
X0
→
(
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
(
(
(
(
(
atleast6
∅
∧
(
(
SNo_
(
Inj1
X6
)
∅
∧
(
¬
exactly2
X1
)
)
→
(
(
atleast2
X4
→
(
¬
atleast2
(
ordsucc
X0
)
)
→
(
¬
nat_p
(
ap
X5
X6
)
)
)
→
(
(
(
(
(
(
¬
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly3
X5
)
)
∧
(
(
(
(
(
¬
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
nat_p
X5
)
∧
(
(
(
(
(
(
¬
SNoLt
X0
X1
)
→
(
ordinal
X6
∧
(
(
¬
ordinal
X5
)
→
setsum_p
X0
)
)
→
(
(
(
(
¬
atleast2
X2
)
→
(
¬
atleast3
X0
)
)
∧
(
(
(
(
(
¬
TransSet
X6
)
∧
(
(
(
atleast2
X6
→
(
¬
atleast3
X1
)
)
∧
(
(
¬
setsum_p
∅
)
→
(
¬
(
X5
∈
X6
)
)
)
)
→
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
(
X2
⊆
X8
)
)
)
)
)
)
→
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
(
(
(
(
¬
PNoLe
X0
(
λX7 :
set
⇒
(
¬
exactly5
X3
)
→
exactly3
X0
)
X5
(
λX7 :
set
⇒
(
(
atleast3
X6
∧
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
atleast4
X6
)
)
)
→
(
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
→
(
(
¬
atleast4
X6
)
∧
(
¬
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
∈
X2
)
)
)
)
→
(
tuple_p
X6
X0
→
(
¬
atleast5
X2
)
)
→
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
¬
atleast2
X0
)
)
→
(
exactly2
X2
∧
(
(
(
(
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast4
X5
)
)
→
(
¬
atleast4
X6
)
)
∧
(
TransSet
(
Inj0
X5
)
→
(
X3
∈
X2
)
)
)
→
set_of_pairs
X5
)
→
atleast5
X5
)
)
)
∧
(
(
(
exactly4
X3
→
(
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
(
X3
∈
X2
)
)
)
→
TransSet
(
V_
X5
)
)
→
atleast2
X5
)
→
(
(
¬
atleast4
X6
)
∧
atleast6
X2
)
)
)
)
→
(
(
¬
SNoLt
X3
X6
)
∧
(
(
¬
bij
X6
X6
(
λX7 :
set
⇒
∅
)
)
∧
(
¬
(
Sing
X6
⊆
X5
)
)
)
)
)
)
∧
(
(
¬
(
X5
⊆
X4
)
)
∧
(
¬
atleast4
X0
)
)
)
)
∧
(
(
¬
set_of_pairs
X6
)
∧
(
(
atleast2
X0
∧
TransSet
X0
)
∧
nat_p
X5
)
)
)
→
(
SNo
X6
∧
(
(
(
¬
ordinal
X6
)
∧
ordinal
X2
)
→
(
¬
atleast3
X6
)
→
(
(
¬
exactly2
X3
)
∧
(
(
(
¬
TransSet
X2
)
→
(
¬
exactly3
X5
)
→
(
¬
exactly5
X4
)
)
∧
(
(
¬
atleast5
X6
)
→
atleast2
X6
)
)
)
→
(
¬
atleast3
X6
)
)
)
)
→
(
(
(
¬
exactly5
X6
)
→
(
¬
atleast3
X5
)
)
∧
(
¬
atleast5
X5
)
)
)
→
(
(
(
(
(
¬
atleast6
X5
)
→
(
(
(
¬
setsum_p
X5
)
∧
(
(
¬
atleast4
∅
)
∧
(
¬
atleast5
X6
)
)
)
∧
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
exactly4
X1
∧
(
(
TransSet
X8
∧
atleast4
(
Sep
(
Inj1
X1
)
(
λX9 :
set
⇒
exactly2
X9
)
)
)
∧
tuple_p
∅
X8
)
)
∧
(
(
(
¬
exactly4
X3
)
∧
(
¬
atleast6
X7
)
)
∧
(
(
¬
atleast5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast5
X0
→
set_of_pairs
X8
→
(
SNo
(
Sing
X8
)
∧
(
(
atleast5
X2
→
atleast4
X7
)
∧
(
¬
(
X6
∈
X8
)
)
)
)
)
)
)
→
(
¬
atleast4
X8
)
)
)
)
∧
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
exactly2
X5
∧
(
(
¬
ordinal
∅
)
→
(
set_of_pairs
X0
∧
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
(
¬
PNoLe
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
(
λX7 :
set
⇒
setsum_p
X6
)
X6
(
λX7 :
set
⇒
(
(
¬
setsum_p
X7
)
∧
SNo
X5
)
)
)
∧
(
¬
atleast4
X5
)
)
∧
(
(
atleast2
X4
→
(
exactly4
X6
∧
SNo
X0
)
)
→
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
(
(
¬
atleast3
X5
)
∧
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast4
X3
)
)
∧
nat_p
X5
)
)
)
→
(
¬
(
X0
∈
X5
)
)
)
)
)
)
→
exactly5
(
⋃
∅
)
)
→
(
¬
exactly4
X3
)
)
)
)
)
)
∧
(
¬
atleast2
X5
)
)
→
atleast3
X0
)
→
(
(
¬
exactly4
X6
)
∧
SNo
X6
)
)
)
→
(
¬
exactly3
X5
)
)
∧
(
¬
TransSet
X6
)
)
∧
(
¬
ordinal
X0
)
)
)
→
(
(
¬
set_of_pairs
∅
)
∧
(
exactly2
X4
→
exactly3
X5
)
)
)
)
∧
(
¬
SNo
X6
)
)
→
atleast5
X6
→
SNo
(
𝒫
X6
)
→
(
(
¬
setsum_p
X5
)
∧
(
(
atleast5
X5
→
(
exactly4
X5
∧
(
(
(
¬
exactly2
X6
)
→
(
(
(
(
(
(
(
¬
exactly3
X4
)
→
exactly3
X5
)
→
(
(
(
atleast3
X5
∧
(
(
¬
nat_p
X5
)
∧
(
¬
SNo
X6
)
)
)
∧
(
¬
ordinal
X5
)
)
∧
atleast5
X5
)
)
→
exactly5
X1
→
(
¬
set_of_pairs
X5
)
)
∧
(
(
(
exactly2
X0
∧
(
atleast2
X0
∧
(
(
exactly4
(
𝒫
X6
)
∧
(
(
(
(
(
atleast6
X5
→
(
TransSet
X6
∧
(
(
¬
exactly2
(
Unj
X5
)
)
∧
(
¬
SNo_
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
→
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
TransSet
(
UPair
X3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
equip
X5
X6
)
∧
(
(
(
exactly3
X5
→
(
¬
atleast3
(
ordsucc
X6
)
)
)
∧
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
(
(
(
atleast3
X6
→
(
¬
atleast6
X6
)
)
→
atleast6
X5
)
→
(
(
(
¬
atleast6
(
ordsucc
X5
)
)
→
ordinal
X5
)
∧
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
∈
X0
)
)
)
)
)
)
∧
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
∧
(
¬
atleast4
X1
)
)
∧
ordinal
X5
)
)
∧
(
¬
exactly5
X5
)
)
→
set_of_pairs
X6
)
→
(
(
(
atleast6
X0
∧
(
(
atleastp
X0
X4
→
(
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
¬
exactly4
X5
)
→
(
(
(
atleast4
X6
→
setsum_p
X5
)
→
(
(
¬
atleastp
X5
X5
)
∧
(
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
→
atleast4
X6
)
)
)
∧
(
¬
ordinal
(
Inj0
X5
)
)
)
)
)
)
→
(
¬
atleast5
X5
)
)
)
∧
(
¬
exactly2
(
V_
X1
)
)
)
∧
(
(
¬
ordinal
X6
)
∧
(
(
(
(
(
(
(
¬
atleast6
X6
)
→
(
nat_p
X3
∧
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast5
X7
)
∧
(
(
(
¬
nat_p
X0
)
∧
exactly4
X5
)
→
ordinal
(
proj0
X6
)
)
)
)
)
∧
(
(
(
¬
(
X0
=
X4
)
)
→
nat_p
X6
)
∧
atleast3
X5
)
)
→
(
SNo_
X6
X0
∧
(
¬
atleast6
X5
)
)
→
(
exactly4
X6
∧
(
(
¬
atleast3
X0
)
→
(
X6
⊆
Sing
X0
)
)
)
→
(
¬
exactly5
X5
)
)
∧
(
X1
=
X6
)
)
→
(
(
(
(
¬
ordinal
X6
)
∧
(
(
exactly2
X0
∧
(
irreflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast4
X0
)
)
∧
(
(
¬
exactly2
X3
)
∧
(
(
¬
atleast6
(
mul_nat
X6
X1
)
)
∧
atleast5
X2
)
)
)
)
∧
(
(
exactly2
∅
∧
(
(
(
(
(
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
(
¬
equip
X6
(
𝒫
X5
)
)
)
∧
(
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
SNo
X5
)
∧
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
exactly4
X6
∧
atleast4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
(
¬
exactly3
X5
)
)
→
TransSet
X6
)
→
(
¬
exactly4
X0
)
)
)
∧
exactly2
X6
)
)
)
∧
(
¬
exactly5
(
lam
X5
(
λX7 :
set
⇒
X6
)
)
)
)
∧
(
(
¬
atleast5
X6
)
∧
(
(
(
¬
ordinal
X5
)
→
(
ordinal
X0
∧
(
¬
atleast6
X0
)
)
)
→
(
(
(
atleast5
X0
→
(
¬
atleast2
∅
)
)
→
exactly1of2
(
(
exactly2
X1
∧
(
¬
exactly4
X6
)
)
→
atleast3
X5
)
(
¬
TransSet
X2
)
)
∧
atleast6
X0
)
)
)
)
)
∧
nat_p
X0
)
)
)
)
∧
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
(
(
(
(
(
¬
SNoEq_
X6
X2
X2
)
∧
(
(
atleast4
X6
∧
(
TransSet
X5
→
(
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
(
¬
atleastp
X7
X8
)
∧
exactly3
X8
)
→
nat_p
∅
)
→
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
ordinal
(
lam
(
SetAdjoin
X8
X7
)
(
λX9 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
(
¬
exactly2
X2
)
∧
(
(
SNo
X6
→
atleast4
X8
)
→
TransSet
X8
)
)
∧
(
atleast2
X7
∧
(
¬
exactly4
∅
)
)
)
∧
(
¬
(
X7
∈
X7
)
)
)
)
→
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
atleast3
X2
∧
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
¬
atleast5
X6
)
)
)
)
∧
(
¬
atleast2
∅
)
)
)
∧
(
PNoEq_
X0
(
λX7 :
set
⇒
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
(
λX7 :
set
⇒
(
(
setsum_p
X7
∧
exactly3
X6
)
∧
exactly4
X6
)
)
→
(
¬
ordinal
X6
)
)
)
→
atleast4
X6
)
∧
PNo_downc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
(
X8
∅
∧
(
¬
TransSet
X7
)
)
→
TransSet
(
Sing
X0
)
)
X6
(
λX7 :
set
⇒
(
(
¬
SNo
(
SNoElts_
X4
)
)
→
(
¬
SNo
X7
)
)
→
(
¬
nat_p
X2
)
→
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∈
X6
)
)
)
∧
(
¬
exactly4
X0
)
)
→
(
(
(
(
¬
ordinal
X5
)
→
(
atleast5
X0
∧
(
(
¬
SNo
X5
)
→
(
¬
exactly3
X4
)
)
)
)
∧
exactly4
X5
)
∧
(
equip
X6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
atleast5
X6
)
)
→
(
(
(
(
(
¬
ordinal
X5
)
→
(
¬
(
X5
∈
X5
)
)
)
→
(
(
¬
atleast6
X0
)
∧
(
(
(
exactly3
X5
∧
(
(
(
¬
TransSet
X6
)
→
exactly2
∅
)
→
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
X7
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
¬
exactly3
X2
)
)
→
(
¬
exactly5
X6
)
)
)
)
∧
(
exactly5
(
Inj0
X5
)
→
atleast3
X1
)
)
∧
atleast5
X5
)
)
)
)
)
→
(
(
(
(
¬
exactly5
X5
)
→
(
¬
exactly5
X0
)
)
→
(
¬
atleast6
X5
)
)
∧
(
PNoLe
X2
(
λX7 :
set
⇒
ordinal
X3
)
(
binintersect
X6
X6
)
(
λX7 :
set
⇒
(
¬
exactly2
X3
)
)
→
(
(
(
(
X0
⊆
X5
)
∧
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast2
X0
)
)
∧
(
(
¬
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
eqreln_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
¬
exactly3
X6
)
∧
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
exactly5
X8
)
→
ordinal
X5
)
∧
(
(
¬
TransSet
X1
)
∧
(
(
(
¬
trichotomous_or_i
(
λX9 :
set
⇒
λX10 :
set
⇒
exactly3
X10
)
)
∧
exactly3
(
V_
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
(
¬
exactly5
X7
)
)
)
)
)
)
)
)
)
→
exactly2
X3
)
)
)
→
(
exactly2
∅
∧
set_of_pairs
X0
)
)
∧
(
(
(
set_of_pairs
X5
→
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
nat_p
X0
)
∧
(
(
¬
exactly2
X6
)
→
(
¬
exactly4
X5
)
)
)
)
→
(
¬
exactly5
X6
)
)
∧
(
¬
nat_p
X5
)
)
)
)
)
→
(
¬
exactly3
X3
)
)
)
→
(
¬
(
X2
⊆
proj1
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
(
atleast3
X0
→
(
(
(
(
¬
atleast4
(
Sing
X2
)
)
∧
(
¬
equip
X3
X1
)
)
∧
(
¬
exactly4
X1
)
)
∧
(
(
(
(
¬
atleast3
X3
)
→
(
atleast5
X0
∧
(
(
atleast4
X2
∧
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
(
(
¬
setsum_p
X3
)
∧
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
PNoLt_
X6
(
λX7 :
set
⇒
(
¬
atleast2
(
proj1
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
(
λX7 :
set
⇒
atleast5
X6
)
→
(
¬
set_of_pairs
X6
)
)
)
∧
set_of_pairs
X3
)
)
)
)
∧
exactly2
X2
)
→
atleast5
X2
)
)
)
→
(
(
¬
TransSet
X3
)
∧
(
(
(
¬
linear_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
TransSet
(
UPair
X4
X4
)
)
∧
exactly5
X2
)
)
)
)
∧
TransSet
X3
)
)
)
)
→
(
∀X3
⊆
∅
,
∀X4 :
set
,
(
¬
exactly4
∅
)
)
)
)
→
(
∃X1 ∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
(
(
∃X2 :
set
,
(
(
∀X3
∈
X1
,
(
¬
exactly4
X2
)
)
∧
(
exactly2
X0
∧
(
¬
exactly4
X1
)
)
)
)
∧
(
∃X2 :
set
,
∀X3 :
set
,
nat_p
∅
)
)
)
In Proofgold the corresponding term root is
b6fe65...
and proposition id is
17a76c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMTZ9eR4pdP1UxeqzQqfN5yv5dXubCdpFRM
)
∀X0
⊆
∅
,
∃X1 ∈
∅
,
∃X2 :
set
,
(
∃X3 :
set
,
(
exactly3
X1
∧
(
∀X4 :
set
,
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast2
X5
)
)
∧
(
TransSet
X3
→
(
(
(
¬
atleast3
X0
)
∧
(
exactly2
X4
→
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
atleast4
X3
∧
(
¬
atleast4
X4
)
)
)
)
)
)
)
)
→
(
¬
atleast2
X0
)
In Proofgold the corresponding term root is
476b31...
and proposition id is
9eb10f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMUUNhEw7Uc26GmeMxvD242MWWKvRHRBHdD
)
∀X0 :
set
,
(
∀X1 :
set
,
(
∃X2 :
set
,
∀X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
nat_p
X3
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
∃X3 :
set
,
∃X4 ∈
Inj1
X0
,
(
¬
atleast3
X4
)
→
(
¬
ordinal
X3
)
)
)
)
)
→
(
∃X1 :
set
,
(
(
∃X2 :
set
,
(
¬
TransSet
X2
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∃X3 :
set
,
(
(
X3
⊆
Unj
X2
)
∧
(
¬
inj
X2
X2
(
λX4 :
set
⇒
X2
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
89185a...
and proposition id is
68566f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYubzBzy5KXVSJKUQf5yopMoVFR7FrjcEm
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
∀X1
∈
X0
,
∃X2 ∈
∅
,
∀X3 :
set
,
∃X4 :
set
,
(
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleastp
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
X0
)
)
→
setsum_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
SNo_
X3
X3
→
(
X0
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
atleast5
X4
)
)
)
In Proofgold the corresponding term root is
c1b81e...
and proposition id is
9c541a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcivZb9qdF25uDhjB6cai2xKzSHyK1BKAf
)
∀X0 :
set
,
∃X1 :
set
,
(
(
∀X2
∈
⋃
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
,
(
¬
exactly3
X2
)
)
∧
(
∀X2 :
set
,
(
∃X3 :
set
,
(
(
∀X4 :
set
,
(
(
¬
binop_on
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X0
)
)
∧
(
(
setsum_p
X4
→
(
X2
=
∅
)
)
∧
set_of_pairs
X0
)
)
)
∧
(
∃X4 :
set
,
(
¬
exactly4
X3
)
)
)
)
→
(
∀X3
⊆
X1
,
∃X4 :
set
,
(
(
¬
(
𝒫
X4
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
¬
SNo_
X3
(
PSNo
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
λX5 :
set
⇒
(
¬
exactly5
X5
)
)
)
)
∧
SNoLt
X4
∅
)
∧
(
(
¬
nat_p
X3
)
∧
atleast5
X4
)
)
)
)
→
(
¬
exactly2
X1
)
)
)
In Proofgold the corresponding term root is
8d37fb...
and proposition id is
327ccd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMLFPXLU1Az5hWtcj7qSWu8Yp7JcnkNfmUN
)
∃X0 ∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
∃X1 :
set
,
(
(
∀X2 :
set
,
(
(
∀X3 :
set
,
∀X4 :
set
,
setsum_p
X3
)
→
exactly3
X2
)
→
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
atleast2
X3
→
(
(
¬
(
X4
∈
X4
)
)
∧
(
(
ordinal
X4
→
(
¬
exactly3
X3
)
)
→
(
(
¬
SNo
X3
)
∧
(
¬
exactly2
X3
)
)
)
)
→
(
¬
exactly3
X2
)
)
→
(
X4
∈
X4
)
)
→
reflexive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
atleast5
X3
→
(
(
¬
(
Inj0
X0
∈
X5
)
)
∧
(
(
¬
atleast5
X2
)
∧
(
(
(
(
(
X0
∈
X5
)
∧
(
(
(
¬
exactly4
X5
)
∧
(
¬
TransSet
X5
)
)
∧
(
atleast2
X4
∧
(
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
¬
exactly4
X4
)
)
∧
exactly5
X4
)
)
)
)
)
∧
(
(
exactly3
X4
∧
(
exactly5
X5
∧
atleast5
X4
)
)
→
(
¬
exactly2
X5
)
)
)
∧
(
setsum_p
X4
→
(
(
(
(
¬
setsum_p
X0
)
→
(
SNo_
∅
∅
∧
(
(
(
SNo
X5
∧
(
(
(
¬
atleast3
X0
)
→
atleast2
X4
)
∧
exactly4
X0
)
)
→
(
¬
atleast4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
¬
TransSet
X6
)
∧
SNoLe
X0
X7
)
)
)
)
)
)
∧
exactly5
X4
)
∧
(
TransSet
X3
∧
(
exactly3
X4
∧
(
(
¬
atleast4
X1
)
∧
(
atleastp
X5
X4
∧
(
¬
SNo
(
⋃
X4
)
)
)
)
)
)
)
)
)
→
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
∀X4
⊆
ordsucc
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
,
ordinal
X1
)
)
)
∧
(
¬
exactly3
X1
)
)
In Proofgold the corresponding term root is
5f76e1...
and proposition id is
7096f5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMQaVHp5UysP9Kb7wPrW64VW8DhbUQfW76G
)
∃X0 ∈
∅
,
∀X1 :
set
,
(
∃X2 :
set
,
(
(
X2
⊆
binrep
X1
X0
)
∧
(
∃X3 ∈
X1
,
(
(
(
∀X4 :
set
,
setsum_p
X3
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X0
)
∧
exactly5
X4
)
)
)
∧
(
(
(
¬
atleast6
(
Inj0
X2
)
)
∧
atleast3
(
add_nat
X3
X3
)
)
∧
PNoEq_
X2
(
λX4 :
set
⇒
(
¬
exactly3
X3
)
)
(
λX4 :
set
⇒
(
(
(
TransSet
X0
→
(
SNoLe
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
X4
∧
(
ordinal
∅
∧
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
tuple_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
X0
)
→
exactly5
X3
)
→
setsum_p
X4
)
)
)
)
)
)
→
(
∃X2 :
set
,
(
¬
atleast4
X2
)
)
In Proofgold the corresponding term root is
5c0912...
and proposition id is
118fcb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMWkY6dRsoLhTt5fQbEeH7aAsJdmrM8k9cY
)
∃X0 :
set
,
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
(
(
∀X3
∈
X1
,
∀X4 :
set
,
(
(
(
¬
atleast6
X4
)
∧
TransSet
X4
)
∧
(
(
¬
atleast3
X4
)
→
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
∧
(
∃X3 :
set
,
(
(
∀X4 :
set
,
(
(
(
(
¬
nat_p
X3
)
→
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
¬
(
X4
∈
binunion
X2
X3
)
)
∧
(
atleast6
X4
∧
(
¬
atleast3
X2
)
)
)
)
∧
(
(
¬
set_of_pairs
X1
)
∧
(
(
¬
(
∅
∈
X3
)
)
∧
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly4
X6
)
)
)
)
∧
(
¬
atleast2
X3
)
)
)
∧
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
∧
SNo
X0
)
In Proofgold the corresponding term root is
c59c27...
and proposition id is
8b515e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVxpDGAsUSPS8moMMSmXJ9m9BdSBLwWuqp
)
∃X0 ∈
∅
,
∀X1 :
set
,
(
reflexive_i
(
λX2 :
set
⇒
λX3 :
set
⇒
∃X4 ∈
X1
,
atleast2
X4
→
tuple_p
X3
X4
)
→
(
∀X2
⊆
X1
,
SNo
(
setexp
X0
X2
)
)
)
→
(
∃X2 :
set
,
ordinal
X2
)
In Proofgold the corresponding term root is
8929e0...
and proposition id is
61c9bc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMT451y3yTGWfK6ukqE1VeKDywp5dU5iiQv
)
∃X0 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
,
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
∃X2 :
set
,
(
(
∀X3
∈
X0
,
atleast6
X2
)
∧
(
(
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
∀X4
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
PNoEq_
X3
(
λX5 :
set
⇒
(
(
¬
atleast5
X5
)
∧
(
¬
TransSet
X4
)
)
)
(
λX5 :
set
⇒
(
exactly5
X4
∧
exactly3
X2
)
)
)
)
)
→
(
¬
exactly3
X0
)
)
∧
(
∃X3 :
set
,
exactly5
∅
)
)
)
)
)
In Proofgold the corresponding term root is
905131...
and proposition id is
899946...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMTZG17v1bXE28L1Gk2EqTXonhFst9ZLh4w
)
∀X0
⊆
∅
,
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
∀X3 :
set
,
(
¬
reflexive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
ordinal
X0
)
)
)
)
)
)
→
(
(
(
¬
setsum_p
X1
)
∧
(
(
∃X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
(
¬
atleast3
X0
)
)
)
∧
(
(
∃X3 :
set
,
(
(
(
∃X4 :
set
,
(
(
¬
SNoLt
X2
X0
)
→
SNo
X3
)
→
atleast5
X4
)
→
(
∀X4
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
exactly4
X4
)
)
∧
(
¬
atleast2
X3
)
)
)
→
atleast2
(
ordsucc
X2
)
)
)
)
∧
(
atleast3
X1
∧
(
∀X3
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
SNoLt
X3
X3
)
)
)
→
(
∃X3 ∈
X1
,
(
(
∀X4
∈
V_
X2
,
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
(
¬
nat_p
(
V_
X1
)
)
∧
(
∀X4
⊆
X0
,
(
inj
X3
X3
(
λX5 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
¬
atleast3
X3
)
)
)
)
)
)
)
)
)
∧
(
exactly2
X1
→
(
∃X2 :
set
,
∀X3
⊆
X1
,
atleast5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
In Proofgold the corresponding term root is
7764c5...
and proposition id is
7b89ea...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMSJRsr4CoZkng6idgTxXo2gYf2rLWG1YJk
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
∀X1 :
set
,
∃X2 ∈
X0
,
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
(
∀X4
⊆
∅
,
(
¬
SNo
X1
)
→
atleast4
X2
)
∧
(
∀X4 :
set
,
(
¬
exactly2
X2
)
→
(
(
(
¬
atleast5
X4
)
∧
(
atleast6
X3
∧
(
(
(
SNo
X4
→
(
(
(
SNo
X1
∧
(
SNo
X3
→
atleast6
X4
)
)
∧
atleast4
X4
)
∧
(
¬
set_of_pairs
X1
)
)
)
∧
(
¬
atleast6
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
atleast2
X1
∧
(
¬
set_of_pairs
X3
)
)
)
)
)
→
atleast5
X0
)
→
(
(
¬
SNoLt
X3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
(
(
(
¬
atleast6
X2
)
→
(
(
¬
nat_p
X4
)
∧
(
¬
atleast3
X4
)
)
)
∧
(
¬
exactly3
X3
)
)
→
(
(
(
(
TransSet
X4
∧
(
(
setsum_p
X3
∧
(
¬
exactly3
X3
)
)
→
(
(
¬
nat_p
X4
)
∧
atleast2
X1
)
)
)
→
atleast2
X4
)
∧
(
¬
exactly2
X3
)
)
∧
TransSet
X3
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b995aa...
and proposition id is
566e60...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMHJMz9PpCsDTXQuMF1K5FADTCje36UrAAJ
)
∃X0 :
set
,
(
(
∀X1
∈
X0
,
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
TransSet
X1
)
)
∧
(
∃X1 :
set
,
∀X2
⊆
X1
,
(
(
∃X3 ∈
X2
,
(
¬
SNoLt
X3
X1
)
)
∧
(
∀X3 :
set
,
∀X4 :
set
,
(
¬
exactly3
X4
)
→
nat_p
X2
)
)
)
)
In Proofgold the corresponding term root is
01b198...
and proposition id is
a6cf38...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMBtFBLRvMeyw4cpVaa4rVDkgvuVWYBk4r
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
(
¬
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
∈
lam2
X0
(
λX2 :
set
⇒
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
λX2 :
set
⇒
λX3 :
set
⇒
X3
)
)
)
∧
(
∃X2 ∈
X1
,
∀X3 :
set
,
(
∃X4 :
set
,
(
(
X4
⊆
X0
)
∧
PNoLe
X4
(
λX5 :
set
⇒
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
(
exactly2
∅
∧
(
¬
equip
X3
X4
)
)
)
)
)
→
(
∃X4 :
set
,
atleast6
X3
)
→
(
(
∀X4
∈
X3
,
(
¬
(
X4
∈
X3
)
)
)
∧
SNo
X0
)
)
)
)
∧
(
∀X1 :
set
,
(
¬
exactly5
X1
)
→
nat_p
X1
→
(
(
∃X2 :
set
,
(
(
¬
atleast5
∅
)
∧
(
¬
nat_p
X0
)
)
)
∧
(
∃X2 ∈
X0
,
∀X3
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
,
atleast2
X2
)
)
)
)
In Proofgold the corresponding term root is
602636...
and proposition id is
368449...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMGW3BcWwU7eBfGyVraeyDKZfgR1KZ61Mbg
)
∀X0 :
set
,
(
∀X1
⊆
X0
,
∀X2 :
set
,
exactly1of2
(
∃X3 :
set
,
∃X4 :
set
,
(
¬
ordinal
X3
)
)
(
(
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
∀X3 :
set
,
(
∀X4
⊆
SNoLev
X1
,
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
∀X4
⊆
X0
,
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
bij
X1
X0
(
λX4 :
set
⇒
X3
)
)
→
(
∃X4 :
set
,
(
(
¬
TransSet
X2
)
∧
exactly4
X3
)
)
)
)
∧
(
∀X3
⊆
X2
,
∀X4 :
set
,
(
¬
exactly5
X4
)
)
)
)
→
(
∀X1
∈
∅
,
(
¬
exactly4
X0
)
)
In Proofgold the corresponding term root is
4f0eb3...
and proposition id is
5a8574...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMhjwr2JcvTMGYfZD2tv5DH25DPRcJ2vmg
)
∃X0 :
set
,
(
(
∀X1
∈
X0
,
(
∃X2 :
set
,
∀X3 :
set
,
∃X4 ∈
X3
,
(
(
(
¬
atleast6
X3
)
→
(
¬
exactly2
X4
)
)
∧
(
¬
TransSet
X4
)
)
)
→
atleast2
X1
)
∧
(
∀X1
∈
V_
X0
,
(
∀X2 :
set
,
∀X3
∈
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
∃X4 :
set
,
(
(
¬
atleast5
∅
)
∧
exactly2
X2
)
)
→
(
∃X2 ∈
X0
,
atleast3
X1
→
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
TransSet
X1
∧
(
(
(
(
(
(
¬
SNo
X4
)
→
(
(
(
¬
exactly4
X3
)
→
(
¬
exactly3
X3
)
)
∧
(
¬
equip
X4
X3
)
)
→
exactly2
X3
)
→
(
¬
atleast2
X3
)
→
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
(
(
(
tuple_p
X1
∅
→
exactly4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
setsum_p
X1
)
∧
(
¬
setsum_p
X2
)
)
)
→
atleast6
∅
)
→
atleast4
(
SetAdjoin
X3
X4
)
)
→
(
¬
exactly4
X3
)
→
(
(
¬
ordinal
X1
)
∧
atleast4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
TransSet
X3
∧
(
¬
atleast2
X4
)
)
)
∧
(
∃X4 :
set
,
(
(
(
atleastp
(
binunion
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
X4
)
X1
→
(
(
(
(
¬
ordinal
(
Unj
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
exactly3
X3
)
)
∧
(
SNo_
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
binintersect
X4
∅
)
∧
(
¬
SNoEq_
X3
X3
X3
)
)
)
∧
setsum_p
X3
)
)
→
(
TransSet
X3
∧
(
¬
SNo
X4
)
)
→
(
(
atleast3
X1
∧
(
(
(
(
(
¬
nat_p
X4
)
∧
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
(
(
(
(
(
exactly3
X4
→
(
atleast3
X4
∧
(
(
atleast6
X0
→
(
(
¬
TransSet
X1
)
∧
(
X3
∈
X2
)
)
)
→
(
SNoLe
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
X4
∧
(
¬
SNo_
X0
∅
)
)
)
)
)
∧
(
(
(
(
¬
atleast6
X3
)
∧
exactly2
X4
)
∧
(
¬
atleast6
∅
)
)
∧
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
ordinal
X3
)
→
(
(
¬
atleast3
X3
)
∧
(
¬
exactly5
X1
)
)
)
∧
(
atleast2
(
UPair
X0
X4
)
∧
(
¬
exactly2
X3
)
)
)
)
→
(
¬
atleast2
X3
)
)
∧
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
(
exactly2
X0
→
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
atleast5
X2
)
)
)
∧
(
(
(
¬
atleast6
(
Pi
X1
(
λX5 :
set
⇒
X4
)
)
)
∧
(
(
¬
exactly3
∅
)
∧
(
(
¬
SNo_
X3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
(
(
(
bij
X3
X3
(
λX5 :
set
⇒
∅
)
→
(
atleast2
X3
∧
SNoLt
(
SNoElts_
X4
)
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
exactly3
X3
)
∧
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
(
(
¬
ordinal
∅
)
∧
atleast6
X4
)
→
(
¬
atleast5
X4
)
)
)
)
)
∧
(
(
¬
(
X3
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
¬
exactly1of2
(
exactly4
(
Inj1
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
(
¬
atleast6
X4
)
)
∧
atleast4
X3
)
)
)
)
→
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b241a2...
and proposition id is
64f3cf...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYacgtjrbxaJTMWj6C2N1oAph39VLxkKRk
)
∀X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
∀X1
∈
X0
,
(
(
∃X2 ∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
(
∀X4 :
set
,
(
X4
∈
X2
)
)
→
(
¬
exactly3
X0
)
)
)
)
∧
(
∃X2 ∈
X0
,
∀X3 :
set
,
atleast5
X3
)
)
In Proofgold the corresponding term root is
d2fcfd...
and proposition id is
ff1015...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMPYwQEDGi9an9uvnxtgki1X4MugSoKW2nt
)
∃X0 ∈
∅
,
∀X1
∈
combine_funcs
X0
(
PSNo
X0
(
λX2 :
set
⇒
∀X3
∈
X2
,
(
(
X2
∈
X2
)
∧
(
(
(
¬
(
X2
=
X0
)
)
→
(
∀X4
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
,
(
∀X5
∈
X3
,
(
(
(
¬
nat_p
X3
)
∧
SNo
X5
)
∧
(
(
(
¬
nat_p
X4
)
∧
(
(
(
¬
exactly4
X5
)
∧
(
(
(
(
exactly3
X3
∧
exactly2
X4
)
→
(
¬
atleast6
∅
)
→
TransSet
X5
)
∧
(
exactly3
X5
∧
(
(
(
¬
TransSet
X3
)
∧
(
¬
atleast4
X5
)
)
∧
exactly3
∅
)
)
)
→
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
exactly5
X5
)
)
∧
(
(
(
atleast5
X5
∧
(
(
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
setsum_p
X0
∧
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
(
(
(
(
¬
atleast5
X6
)
→
(
(
¬
atleast6
X3
)
∧
(
(
(
exactly4
X6
→
nat_p
X6
)
→
(
(
(
ordinal
X6
∧
(
(
(
¬
(
X6
∈
∅
)
)
→
(
¬
ordinal
X7
)
→
ordinal
X6
→
(
¬
exactly5
X6
)
)
→
(
ordinal
X6
∧
exactly3
(
mul_nat
X7
X7
)
)
)
)
→
atleast3
X6
)
∧
(
(
set_of_pairs
X7
→
(
¬
exactly3
X4
)
)
→
(
¬
linear_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
TransSet
X3
)
)
)
→
exactly3
∅
)
)
)
∧
exactly5
X6
)
)
)
→
(
atleast2
X0
∧
exactly3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
∧
atleast2
X7
)
→
(
¬
atleast5
X0
)
)
∧
(
¬
exactly5
X6
)
)
)
)
)
)
→
atleast3
X5
)
→
(
¬
atleast5
X2
)
)
→
(
(
(
(
¬
exactly5
(
Sing
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
¬
exactly5
X3
)
)
→
(
(
SNo
X5
∧
(
(
¬
exactly5
X5
)
∧
(
(
(
¬
atleast2
X0
)
→
(
ordinal
∅
→
atleast3
∅
)
→
(
¬
nat_p
X5
)
→
(
¬
exactly4
(
⋃
∅
)
)
→
(
(
¬
TransSet
X3
)
∧
(
(
¬
atleast6
X5
)
→
(
PNo_downc
(
λX6 :
set
⇒
λX7 :
set
→
prop
⇒
X7
X5
)
X4
(
λX6 :
set
⇒
(
¬
atleast3
X5
)
)
∧
(
(
atleast6
X3
∧
(
¬
equip
X3
X4
)
)
→
(
(
TransSet
X4
→
set_of_pairs
X4
→
(
(
(
(
¬
ordinal
∅
)
→
atleast4
X4
→
(
¬
atleast3
X5
)
)
→
(
¬
exactly3
X0
)
→
(
(
¬
nat_p
X5
)
∧
(
(
(
atleast4
X5
→
(
¬
setsum_p
X2
)
)
→
(
¬
atleast6
X4
)
)
∧
(
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
nat_p
X5
)
)
)
)
→
(
setsum_p
X4
→
(
¬
exactly5
X5
)
)
→
atleast4
X5
)
→
(
exactly4
X5
∧
(
(
¬
SNo
X5
)
→
(
¬
TransSet
X5
)
)
)
)
∧
(
atleast3
X0
∧
atleast3
X3
)
)
)
)
)
)
)
∧
(
¬
exactly4
X0
)
)
)
)
∧
ordinal
X4
)
)
∧
(
¬
atleast3
X3
)
)
)
)
∧
(
(
atleast4
X4
→
atleast3
X5
)
→
setsum_p
X0
)
)
∧
ordinal
X5
)
)
)
→
(
¬
ordinal
X3
)
)
→
(
∃X5 :
set
,
(
(
X5
⊆
X3
)
∧
(
atleast3
X5
∧
exactly5
X3
)
)
)
→
TransSet
X3
)
)
→
(
∀X4 :
set
,
(
∀X5 :
set
,
(
¬
atleast5
X5
)
→
(
(
(
atleast4
X5
∧
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
(
(
(
atleast3
X5
∧
(
(
¬
exactly4
X6
)
∧
(
(
(
¬
exactly5
X0
)
→
atleast6
X6
→
exactly2
X3
→
(
¬
atleast3
X0
)
)
→
(
atleast5
X0
→
exactly3
(
proj1
∅
)
)
→
(
¬
nat_p
X6
)
)
)
)
∧
(
(
X6
=
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
atleast4
∅
)
)
∧
nat_p
X4
)
∧
(
¬
atleast3
∅
)
)
∧
atleast6
X0
)
)
)
∧
(
(
¬
(
X4
⊆
V_
X5
)
)
∧
SNo
(
Sing
X3
)
)
)
)
∧
(
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
exactly4
X2
)
)
→
atleast6
X2
)
)
)
)
)
(
λX2 :
set
⇒
X2
)
(
λX2 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
,
atleast5
X0
→
TransSet
X0
In Proofgold the corresponding term root is
de1da5...
and proposition id is
d64595...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMKHkHkSRJpDgTWxXS8DYRFdVMgLAcoXc93
)
∃X0 :
set
,
(
(
∃X1 ∈
proj1
∅
,
∀X2
∈
X1
,
∀X3
∈
X0
,
∀X4 :
set
,
(
(
(
atleast3
X3
∧
(
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
nat_p
X3
∧
(
¬
atleast5
X4
)
)
→
(
(
(
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
set_of_pairs
X0
→
(
(
(
(
exactly3
X3
→
atleast2
X2
)
∧
(
¬
atleast3
X3
)
)
→
(
exactly2
X4
∧
(
(
(
¬
atleast4
X3
)
∧
(
¬
SNoLt
X1
X3
)
)
→
(
¬
SNo
X0
)
→
(
(
(
(
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
exactly3
X2
→
(
¬
setsum_p
X3
)
)
→
(
(
(
¬
tuple_p
X4
X3
)
→
atleast4
X2
)
∧
(
(
(
¬
SNo
X4
)
∧
(
¬
TransSet
(
V_
X4
)
)
)
∧
equip
X4
X2
)
)
)
∧
(
(
(
(
(
(
(
(
(
¬
TransSet
X3
)
∧
bij
X3
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
(
λX5 :
set
⇒
X4
)
)
→
(
(
(
¬
(
X4
∈
X2
)
)
∧
atleast4
∅
)
→
TransSet
X4
)
→
(
¬
exactly5
X3
)
)
∧
(
¬
atleast5
∅
)
)
∧
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
(
¬
exactly5
X3
)
→
(
¬
exactly4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
SNo
X2
)
→
(
(
(
(
¬
atleast4
X2
)
∧
(
¬
nat_p
X0
)
)
→
ordinal
X2
)
∧
exactly4
X3
)
)
→
(
exactly2
∅
∧
(
atleast5
X3
∧
(
(
¬
(
X3
∈
⋃
(
Sing
∅
)
)
)
→
(
¬
(
X3
∈
X1
)
)
)
)
)
)
→
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
=
X4
)
→
(
¬
setsum_p
X4
)
)
)
∧
TransSet
X3
)
)
)
)
∧
(
atleast3
∅
∧
(
(
atleast6
X4
→
atleast3
X2
→
atleast3
X0
)
∧
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly4
X6
→
(
¬
ordinal
X6
)
→
nat_p
X5
)
)
∧
(
¬
nat_p
X0
)
)
)
)
)
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
atleast2
X4
)
)
)
∧
atleast2
X4
)
→
nat_p
X2
)
→
(
¬
SNoLe
X0
X2
)
)
∧
(
∀X1 :
set
,
(
¬
atleast4
X0
)
→
(
∃X2 :
set
,
(
(
(
∃X3 ∈
binrep
X0
X1
,
(
(
∃X4 ∈
X3
,
(
TransSet
X3
∧
(
(
¬
SNoLt
X0
X2
)
→
(
atleast6
X0
∧
nat_p
X3
)
→
(
¬
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
)
∧
(
(
∀X4
⊆
∅
,
nat_p
∅
→
(
¬
atleast3
X4
)
→
(
(
¬
exactly3
X4
)
∧
(
(
(
(
(
¬
exactly2
X4
)
→
(
¬
atleast2
X2
)
)
→
(
¬
atleast6
X4
)
→
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast3
X0
)
→
(
ordinal
X2
∧
(
ordinal
X3
∧
(
(
(
¬
atleast5
X4
)
→
ordinal
X4
)
∧
(
(
¬
exactly2
∅
)
∧
(
¬
ordinal
X0
)
)
)
)
)
)
)
)
∧
(
¬
eqreln_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
PNoLe
X5
(
λX6 :
set
⇒
(
SNoLe
X0
X5
→
(
¬
setsum_p
X0
)
)
→
exactly5
X5
)
X5
(
λX6 :
set
⇒
(
(
¬
exactly5
X0
)
∧
(
TransSet
X6
∧
(
¬
exactly4
X5
)
)
)
)
→
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
X0
∈
X6
)
)
)
∧
(
¬
exactly2
∅
)
)
)
→
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
(
(
(
(
(
atleast3
X1
→
(
¬
atleast5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
ordinal
X1
→
atleast6
X5
)
∧
atleast3
X0
)
∧
(
(
(
¬
exactly5
X5
)
∧
(
setsum_p
X4
→
TransSet
X5
)
)
→
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
¬
atleast4
(
Inj0
X0
)
)
)
→
(
(
¬
TransSet
X5
)
∧
(
¬
tuple_p
∅
(
ap
∅
X5
)
)
)
)
∧
(
(
¬
exactly5
X0
)
→
exactly2
X1
)
)
→
(
(
(
(
(
exactly4
X5
∧
(
¬
atleast4
X5
)
)
→
(
(
¬
atleast6
X4
)
∧
exactly3
X4
)
)
→
(
(
(
(
set_of_pairs
X0
→
(
¬
atleast5
X4
)
)
→
(
(
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleastp
∅
X4
)
→
set_of_pairs
X3
)
∧
(
¬
atleast3
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
¬
atleast6
(
SNoElts_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
(
(
atleast3
X2
→
(
(
(
¬
atleast4
(
Sing
X4
)
)
∧
(
¬
atleast6
X4
)
)
∧
(
(
¬
exactly2
X4
)
∧
(
¬
ordinal
X2
)
)
)
)
∧
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
atleast6
X6
)
→
atleast6
X6
)
)
→
(
(
¬
atleast3
X4
)
→
(
(
(
(
(
(
¬
tuple_p
X5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
set_of_pairs
X4
→
exactly3
X2
)
)
→
(
atleast2
∅
∧
(
(
(
¬
exactly4
X2
)
→
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
¬
atleast5
X0
)
)
→
TransSet
X3
)
∧
(
(
nat_p
X2
→
(
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
TransSet
(
V_
X4
)
)
)
)
→
exactly3
X4
)
)
)
)
∧
(
¬
exactly5
X5
)
)
∧
(
¬
TransSet
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
atleast2
(
SNoElts_
X4
)
)
)
→
(
TransSet
X5
∧
(
(
(
(
(
(
(
(
(
¬
ordinal
∅
)
∧
(
set_of_pairs
(
proj1
X5
)
∧
(
¬
atleast2
X5
)
)
)
→
(
¬
atleast5
X3
)
)
→
(
¬
(
X0
∈
X5
)
)
)
∧
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
¬
nat_p
X5
)
∧
(
SNo_
X4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
setsum_p
X4
→
(
¬
ordinal
X4
)
)
)
)
)
→
(
atleast5
X5
→
(
(
¬
exactly4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
atleast4
X1
)
)
→
exactly3
X5
)
∧
atleast3
X4
)
∧
(
¬
atleast3
X4
)
)
)
)
)
→
(
(
(
(
(
¬
exactly2
X5
)
∧
(
(
(
TransSet
X3
→
(
(
¬
exactly3
∅
)
∧
(
(
exactly4
X5
→
(
¬
atleast3
X3
)
)
∧
(
nat_p
(
ordsucc
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast4
X3
)
)
)
)
→
(
(
atleast4
X0
∧
(
setsum_p
X4
→
ordinal
(
Sing
X5
)
)
)
∧
(
(
¬
exactly2
X4
)
∧
(
(
¬
TransSet
X4
)
∧
(
¬
exactly4
X1
)
)
)
)
)
∧
(
(
(
(
(
(
¬
exactly5
X5
)
∧
(
exactly3
X4
∧
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
exactly4
X5
)
∧
(
¬
atleast2
(
Sing
X5
)
)
)
)
∧
(
(
(
exactly4
X0
∧
(
¬
TransSet
X5
)
)
→
(
(
(
(
(
¬
bij
X5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
(
λX6 :
set
⇒
X6
)
)
∧
exactly5
X4
)
→
(
¬
exactly4
X5
)
)
∧
(
(
¬
exactly3
X3
)
→
(
(
(
exactly4
X0
→
atleast4
X4
)
→
(
(
(
¬
nat_p
X0
)
∧
(
exactly4
X4
→
exactly4
X4
)
)
∧
(
¬
atleast5
X5
)
)
)
∧
(
(
¬
exactly3
X4
)
∧
TransSet
X5
)
)
)
)
∧
(
¬
exactly5
X0
)
)
→
(
(
¬
totalorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
(
(
¬
atleast2
X7
)
→
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
¬
TransSet
X3
)
→
atleast2
X6
)
)
∧
(
¬
atleast3
X7
)
)
)
)
∧
(
(
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
exactly3
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
atleast3
∅
)
)
)
)
→
(
(
¬
SNoLe
X5
X5
)
∧
(
¬
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
exactly4
X5
→
(
¬
nat_p
X3
)
)
∧
(
¬
atleast3
X4
)
)
)
)
→
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
atleast3
X4
)
∧
(
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
PNoLe
X7
(
λX8 :
set
⇒
(
¬
exactly4
X7
)
)
X2
(
λX8 :
set
⇒
(
atleast4
X7
∧
(
(
(
(
(
¬
exactly5
X2
)
→
(
(
(
¬
setsum_p
X0
)
∧
(
TransSet
X2
→
ordinal
X6
→
(
(
(
¬
atleast5
(
Inj0
X8
)
)
∧
(
(
atleast2
(
Sing
X8
)
∧
(
per_i
(
λX9 :
set
⇒
λX10 :
set
⇒
atleastp
X5
X9
)
→
(
exactly2
X5
∧
(
(
setsum_p
X0
→
set_of_pairs
X8
)
→
(
set_of_pairs
X5
∧
exactly2
X7
)
)
)
)
)
→
(
(
¬
exactly3
X8
)
→
SNo
X2
)
→
(
¬
equip
X8
∅
)
)
)
∧
(
(
¬
exactly4
X0
)
∧
atleast5
X6
)
)
)
)
∧
setsum_p
X8
)
)
∧
(
X3
∈
X7
)
)
∧
exactly4
X8
)
→
PNoLt
∅
(
λX9 :
set
⇒
(
¬
atleast4
∅
)
)
X7
(
λX9 :
set
⇒
(
¬
atleast2
X8
)
)
)
)
)
∧
(
SNoLe
X6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast4
X7
)
)
)
)
→
exactly3
X0
)
)
)
→
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
∈
X2
)
)
)
)
)
→
(
(
(
¬
atleast5
X4
)
→
(
¬
atleast6
X3
)
→
TransSet
X5
)
∧
(
(
¬
SNo
X3
)
→
(
¬
exactly1of3
(
(
set_of_pairs
X5
→
(
¬
setsum_p
∅
)
)
→
(
(
exactly3
X4
→
SNo
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
ordinal
X5
→
atleast5
X4
→
(
(
(
(
(
(
(
¬
exactly3
X3
)
→
exactly3
X4
→
(
(
(
¬
atleast2
X5
)
∧
(
(
¬
exactly4
X4
)
→
exactly2
X4
)
)
∧
(
(
(
¬
atleast4
X0
)
∧
(
¬
tuple_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∅
)
)
∧
(
(
¬
atleast2
X4
)
∧
(
¬
SNo
(
Sing
X4
)
)
)
)
)
→
(
¬
bij
X0
X5
(
λX6 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
nat_p
X5
)
→
(
¬
(
X4
∈
∅
)
)
)
∧
(
(
(
(
(
(
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
(
(
¬
setsum_p
X4
)
∧
(
atleast5
X5
∧
atleast3
X4
)
)
→
atleast4
X3
)
→
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
nat_p
X5
)
)
→
set_of_pairs
X5
→
(
¬
setsum_p
X5
)
)
∧
(
¬
ordinal
X3
)
)
→
atleast6
X2
)
)
→
(
¬
nat_p
X2
)
→
(
(
(
X1
=
X4
)
→
exactly5
X1
)
∧
(
(
atleast6
X2
→
(
atleast2
X4
∧
(
(
¬
(
X5
=
X3
)
)
∧
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
atleast2
X7
)
)
)
)
)
)
→
(
(
(
(
(
¬
exactly2
X5
)
∧
nat_p
X1
)
→
(
¬
atleast3
X0
)
)
→
(
¬
nat_p
X4
)
)
∧
(
(
atleast2
X4
→
(
¬
exactly4
X4
)
→
(
¬
exactly5
X5
)
)
∧
(
set_of_pairs
X5
∧
(
¬
TransSet
X0
)
)
)
)
)
)
)
∧
(
(
(
(
¬
exactly3
X0
)
∧
(
SNoLt
X1
X0
∧
(
¬
atleast2
(
PSNo
(
ordsucc
X4
)
(
λX6 :
set
⇒
(
¬
atleast5
X2
)
→
(
setsum_p
X0
∧
(
¬
exactly5
X6
)
)
)
)
)
)
)
∧
(
(
(
(
(
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
atleast5
X6
)
)
∧
atleast2
∅
)
∧
setsum_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast4
X4
)
→
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X5
)
)
→
(
(
¬
atleast6
X0
)
∧
exactly5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
(
atleast4
X5
∧
(
¬
ordinal
X2
)
)
)
)
)
)
)
(
stricttotalorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
nat_p
∅
)
∧
(
¬
exactly4
X2
)
)
(
¬
ordinal
X4
)
)
)
)
)
∧
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
setsum_p
X3
)
)
)
→
(
atleast5
X4
∧
atleastp
X5
X4
)
)
)
)
)
)
→
(
∀X3
∈
X0
,
∀X4 :
set
,
(
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
¬
atleast4
X4
)
∧
(
atleast5
X4
∧
(
(
¬
atleast5
X4
)
∧
(
(
(
(
(
(
¬
atleast6
X3
)
→
(
(
(
nat_p
X2
∧
(
atleast3
∅
∧
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
(
(
(
(
¬
atleast4
X1
)
→
(
¬
atleast2
X3
)
→
exactly2
X3
)
→
SNoLe
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
(
V_
(
proj1
X4
)
)
)
→
(
(
(
(
TransSet
X2
∧
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
setsum_p
X6
)
)
)
∧
(
(
SNo_
∅
X4
∧
(
nat_p
X3
→
(
(
¬
exactly4
X4
)
∧
atleast3
X4
)
)
)
→
(
¬
nat_p
(
𝒫
X4
)
)
→
(
(
(
(
(
atleast4
X3
∧
(
(
exactly1of3
(
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
(
(
PNo_upc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
X6
X2
)
X2
(
λX5 :
set
⇒
(
(
(
(
¬
atleast3
X4
)
→
(
¬
atleast2
X4
)
)
∧
(
(
¬
ordinal
X5
)
→
(
(
ordinal
X4
∧
(
(
(
(
¬
exactly5
X5
)
→
(
(
(
¬
TransSet
X4
)
∧
(
atleast4
X2
∧
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
exactly3
X5
∧
(
¬
nat_p
X5
)
)
)
)
→
(
(
¬
inj
X5
X5
(
λX6 :
set
⇒
X5
)
)
∧
ordinal
X5
)
)
→
(
(
nat_p
X4
∧
(
SNo
X2
→
(
(
¬
nat_p
X3
)
∧
setsum_p
∅
)
→
SNo
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
exactly4
X5
∧
(
¬
exactly2
X0
)
)
)
)
)
→
(
(
(
¬
exactly2
X4
)
→
(
¬
tuple_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
X3
)
→
(
¬
atleast5
X3
)
)
∧
(
(
(
¬
exactly5
X0
)
→
(
¬
exactly2
X2
)
→
atleast5
X5
)
∧
(
¬
atleast3
X2
)
)
)
→
(
¬
ordinal
X4
)
→
TransSet
X4
)
→
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
(
X5
∈
X4
)
)
)
)
∧
SNo
X5
)
)
∧
(
(
¬
atleast6
X4
)
→
(
(
(
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
atleast5
X0
)
→
(
atleast2
X3
∧
(
PNoLe
X1
(
λX5 :
set
⇒
exactly2
X4
)
X3
(
λX5 :
set
⇒
bij
(
UPair
X1
X5
)
X4
(
λX6 :
set
⇒
X5
)
)
∧
(
¬
(
X3
∈
X4
)
)
)
)
)
∧
(
(
¬
TransSet
X3
)
→
(
¬
exactly3
X2
)
)
)
→
(
¬
atleast3
∅
)
)
)
∧
(
(
X2
⊆
X4
)
→
setsum_p
X4
)
)
(
atleast2
(
ordsucc
X4
)
∧
TransSet
(
lam2
X4
(
λX5 :
set
⇒
X4
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X6
)
)
)
→
(
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
exactly2
X1
)
)
∧
(
(
(
(
¬
TransSet
X3
)
→
exactly5
X4
)
→
(
atleast5
X3
∧
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
X3
=
X4
)
)
)
)
→
(
(
(
(
(
(
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
¬
ordinal
X0
)
→
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
atleast2
X4
∧
(
(
(
¬
exactly5
X3
)
→
(
(
¬
ordinal
X3
)
∧
(
(
(
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
(
(
(
¬
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
(
(
¬
set_of_pairs
∅
)
∧
(
(
(
(
¬
equip
X4
X3
)
∧
(
(
¬
exactly3
X1
)
∧
(
¬
atleast4
X3
)
)
)
∧
(
exactly4
X3
∧
(
(
¬
ordinal
∅
)
∧
(
(
¬
atleast2
X3
)
→
(
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
exactly3
X6
→
atleast2
X2
)
→
(
¬
setsum_p
X6
)
)
→
atleast6
X2
→
(
setsum_p
X5
∧
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
¬
atleast2
X8
)
→
atleast3
X7
)
→
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
SNo
X8
)
)
)
)
)
)
∧
exactly4
(
Sing
X2
)
)
)
)
)
)
→
ordinal
X2
→
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
(
¬
ordinal
X3
)
∧
(
atleast4
X2
∧
atleast4
X4
)
)
→
(
(
¬
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
atleast2
X0
)
)
)
→
(
¬
atleast2
X3
)
)
)
∧
(
(
¬
atleastp
X3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
atleast3
∅
)
)
)
)
∧
nat_p
∅
)
→
(
¬
atleast2
X3
)
)
∧
(
atleast5
X4
→
SNo
X3
)
)
)
∧
exactly5
∅
)
→
(
(
(
¬
SNo
(
SNoLev
X4
)
)
→
(
exactly4
(
SetAdjoin
X0
X0
)
∧
SNo
X3
)
)
∧
(
(
(
ordinal
X3
∧
(
(
atleast4
X2
∧
(
(
¬
TransSet
X4
)
∧
atleast6
X2
)
)
→
(
¬
TransSet
X4
)
)
)
∧
(
¬
atleast3
X2
)
)
∧
(
(
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
TransSet
(
V_
X1
)
→
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
atleast4
X4
)
)
)
)
→
exactly5
X1
)
)
)
→
(
nat_p
X4
∧
(
(
¬
setsum_p
X2
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
PNo_downc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
(
TransSet
X7
∧
(
(
(
¬
setsum_p
(
proj0
X0
)
)
∧
(
(
X8
X7
∧
(
(
¬
X8
X3
)
∧
(
¬
X8
X7
)
)
)
→
(
setsum_p
X7
∧
(
(
(
¬
X8
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
atleast6
(
Inj0
X1
)
)
)
→
(
¬
exactly5
(
𝒫
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
)
∧
(
(
(
(
X8
X5
→
atleast4
X7
)
∧
X8
X7
)
∧
atleast6
X6
)
∧
(
¬
X8
X6
)
)
)
)
)
(
SNoLev
X5
)
(
λX7 :
set
⇒
(
(
(
atleast6
X0
→
atleast5
X1
→
(
(
¬
atleast3
X1
)
∧
exactly4
X7
)
)
∧
(
¬
set_of_pairs
X5
)
)
∧
exactly5
X7
)
)
)
→
(
TransSet
(
mul_nat
(
lam2
X4
(
λX5 :
set
⇒
X0
)
(
λX5 :
set
⇒
λX6 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
X3
)
∧
(
¬
exactly2
X4
)
)
→
(
atleast6
X2
∧
TransSet
X2
)
→
(
(
(
(
(
(
(
(
(
¬
PNoLt
X2
(
λX5 :
set
⇒
(
(
¬
totalorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
strictpartialorder_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
(
SNo
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast4
X5
)
)
→
(
(
exactly4
∅
→
atleast4
X9
)
∧
(
¬
(
X9
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
set_of_pairs
X1
)
→
atleast2
X6
)
→
ordinal
X7
)
)
∧
(
¬
SNo
X0
)
)
)
X3
(
λX5 :
set
⇒
(
atleast6
X5
→
(
(
(
(
nat_p
X4
∧
(
(
¬
set_of_pairs
X4
)
→
(
¬
atleast6
X5
)
)
)
→
(
(
¬
(
∅
∈
X1
)
)
∧
(
(
¬
nat_p
X3
)
→
TransSet
X5
)
)
)
→
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
exactly3
X3
)
)
)
→
(
¬
exactly5
X5
)
)
)
∧
ordinal
X3
)
→
atleast3
X3
→
exactly3
X2
)
→
(
¬
nat_p
X3
)
→
(
exactly4
X0
∧
(
¬
atleast5
X2
)
)
)
∧
(
(
(
¬
exactly2
X4
)
∧
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
ordinal
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
exactly2
X0
→
TransSet
X3
→
(
atleast4
X3
∧
(
(
(
¬
TransSet
X0
)
∧
ordinal
X4
)
∧
(
(
¬
ordinal
X4
)
∧
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
exactly2
X1
→
nat_p
X3
→
(
(
¬
exactly4
X4
)
∧
(
¬
exactly5
X3
)
)
)
→
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
atleast5
X2
∧
(
¬
atleast3
X3
)
)
)
∧
atleastp
X3
X0
)
∧
atleast6
X0
)
)
)
)
)
)
)
→
(
¬
atleast6
X2
)
→
(
(
¬
atleast3
X3
)
∧
(
atleast6
X4
∧
(
(
¬
SNoLt
(
binunion
(
proj1
X0
)
X3
)
X4
)
→
(
¬
exactly4
X4
)
)
)
)
)
→
(
¬
(
X3
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
atleast2
(
Inj1
X2
)
)
∧
(
(
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
atleast5
(
mul_nat
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X3
)
)
→
(
(
(
(
¬
SNo_
X4
∅
)
→
atleast6
X4
)
∧
(
nat_p
∅
→
(
(
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
(
(
(
(
¬
atleast2
X4
)
→
(
¬
nat_p
X2
)
→
TransSet
X0
)
∧
(
(
¬
TransSet
X4
)
∧
(
(
(
¬
exactly2
X2
)
→
(
(
(
¬
exactly4
X0
)
→
exactly2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
inj
X3
X4
(
λX5 :
set
⇒
X5
)
)
)
∧
(
(
SNoLt
X3
X3
→
exactly4
X2
)
∧
nat_p
X3
)
)
)
→
(
(
¬
atleast5
X3
)
∧
(
(
(
(
(
(
atleast3
X3
∧
(
TransSet
X3
∧
(
(
¬
exactly1of3
(
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
(
(
¬
atleast6
(
setprod
(
⋃
X0
)
X3
)
)
→
(
atleast4
X1
∧
exactly5
X0
)
)
(
setsum_p
(
proj1
X3
)
)
)
→
(
¬
TransSet
X4
)
)
)
)
∧
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
SNo
∅
)
)
)
)
∧
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
atleast3
∅
∧
atleast3
X6
)
→
(
¬
ordinal
X6
)
→
(
(
¬
atleast5
X6
)
∧
ordinal
X6
)
)
→
exactly3
X4
)
)
∧
atleast5
X3
)
→
exactly4
X1
)
∧
atleast2
X3
)
)
)
)
)
→
(
¬
exactly3
(
proj0
X2
)
)
)
)
∧
(
(
X2
∈
X3
)
→
(
¬
SNo_
X1
X3
)
)
)
)
)
∧
(
¬
exactly4
X3
)
)
)
)
→
(
(
¬
exactly5
X3
)
∧
(
(
(
(
(
(
TransSet
X3
∧
(
(
(
(
nat_p
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
exactly4
X2
)
)
→
exactly3
X3
→
(
¬
atleast2
X0
)
)
∧
(
¬
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
=
X3
)
)
)
→
(
¬
atleast2
X3
)
→
(
ordinal
X1
→
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
∈
X4
)
)
→
(
(
equip
X3
X4
∧
exactly5
X2
)
∧
(
¬
exactly4
X1
)
)
)
)
∧
(
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
SNoLe
X3
X3
→
atleast5
X3
)
)
)
∧
(
(
atleast6
X4
∧
atleast4
∅
)
∧
exactly2
(
SNoElts_
X3
)
)
)
→
(
(
¬
atleast2
X0
)
∧
(
¬
nat_p
X3
)
)
)
∧
(
(
(
¬
setsum_p
X3
)
∧
atleast3
X3
)
→
(
tuple_p
X2
∅
∧
(
(
(
(
¬
exactly3
X4
)
∧
exactly2
X3
)
∧
(
set_of_pairs
X3
→
(
¬
exactly4
X4
)
)
)
∧
(
¬
TransSet
(
ReplSep
X3
(
λX5 :
set
⇒
(
exactly5
∅
∧
(
¬
exactly3
(
Sing
(
Inj0
X0
)
)
)
)
)
(
λX5 :
set
⇒
X1
)
)
)
)
)
)
)
→
(
SNo
X0
∧
(
exactly2
X2
→
(
¬
exactly5
(
SNoElts_
X2
)
)
)
)
)
)
)
∧
TransSet
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
atleast6
X0
)
→
(
TransSet
X0
∧
(
(
exactly2
X3
→
(
(
¬
ordinal
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
¬
TransSet
X3
)
)
)
→
(
¬
atleast4
X4
)
)
)
)
∧
ordinal
X0
)
)
)
→
(
¬
exactly2
X4
)
)
∧
exactly5
X4
)
)
∧
(
¬
set_of_pairs
(
setminus
∅
X3
)
)
)
→
(
(
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
exactly5
X3
)
)
∧
(
(
atleast5
X1
→
(
(
¬
setsum_p
X3
)
→
atleast3
∅
)
→
(
(
SetAdjoin
(
Sep
X0
(
λX5 :
set
⇒
(
set_of_pairs
X4
∧
atleast5
X0
)
→
(
exactly2
X2
∧
set_of_pairs
(
binunion
∅
∅
)
)
)
)
X2
=
X1
)
∧
ordinal
∅
)
)
→
nat_p
X1
)
)
→
(
¬
ordinal
X1
)
)
)
∧
(
¬
exactly4
X0
)
)
)
→
(
¬
nat_p
X1
)
)
∧
exactly5
X3
)
∧
(
(
(
(
X4
∈
X3
)
→
exactly4
X2
)
→
(
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
TransSet
X5
)
∧
(
(
(
(
¬
atleast2
X4
)
→
(
¬
exactly3
X4
)
)
→
ordinal
X4
)
→
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
binop_on
X5
(
λX7 :
set
⇒
λX8 :
set
⇒
X8
)
∧
(
¬
nat_p
X5
)
)
)
∧
(
(
(
(
(
exactly5
X3
→
setsum_p
∅
)
→
(
¬
exactly2
X4
)
→
(
¬
exactly3
X0
)
)
→
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast5
X5
)
)
∧
(
¬
atleast4
X4
)
)
→
atleast5
X3
→
(
¬
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast5
X3
→
(
¬
atleast2
X4
)
)
)
)
)
)
→
(
(
set_of_pairs
X0
→
(
¬
atleast5
X1
)
→
(
¬
atleast4
X0
)
)
∧
(
(
¬
atleast6
X3
)
→
(
setsum_p
X4
∧
(
¬
(
X2
∈
X3
)
)
)
)
)
)
)
∧
(
atleast3
∅
→
(
¬
exactly5
X3
)
→
(
¬
exactly5
X3
)
)
)
)
)
)
)
→
(
¬
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
∧
(
(
∀X3 :
set
,
(
∀X4 :
set
,
(
(
¬
ordinal
X2
)
∧
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
∃X4 ∈
X1
,
(
¬
atleast2
X1
)
)
)
→
(
∃X3 :
set
,
setsum_p
X1
→
(
∃X4 ∈
X0
,
exactly2
X4
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b8ec53...
and proposition id is
e31e5e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMX4xSmGUNhKQNM3yNyEccZsdzByXU5uhPK
)
∀X0 :
set
,
∀X1 :
set
,
(
∀X2 :
set
,
(
∀X3
∈
Sing
X2
,
(
(
∀X4
∈
X1
,
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
nat_p
∅
∧
atleast3
X4
)
∧
(
¬
atleast3
X4
)
)
)
∧
(
∀X4
∈
X0
,
SNo
X2
→
(
¬
exactly2
X4
)
)
)
)
→
(
¬
transitive_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
¬
exactly4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
∀X2
∈
X0
,
(
¬
atleast3
X1
)
)
In Proofgold the corresponding term root is
7c39df...
and proposition id is
3f67d1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMUVz8Jxzr4H2Ce9V9bzukNJigwUCuLwyG4
)
∃X0 :
set
,
(
(
∀X1
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
(
(
∀X2 :
set
,
∀X3
∈
X0
,
∀X4
⊆
X2
,
(
(
atleast4
X3
→
(
¬
nat_p
X4
)
)
∧
(
(
exactly3
(
⋃
X3
)
∧
exactly3
X4
)
∧
(
(
ordinal
∅
→
(
(
¬
exactly2
X3
)
∧
(
atleast3
X4
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
exactly5
X5
∧
(
(
¬
SNo
X1
)
∧
(
(
(
atleast3
X6
∧
(
¬
TransSet
X0
)
)
→
(
(
(
(
(
(
(
¬
exactly2
X5
)
→
(
¬
atleast2
X2
)
)
→
(
TransSet
X6
∧
(
atleast5
∅
∧
(
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
nat_p
X2
)
→
(
¬
exactly5
X6
)
)
)
)
→
(
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
atleast6
X6
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
→
(
¬
(
X0
=
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
set_of_pairs
X5
)
→
(
¬
SNoLe
X0
X2
)
→
PNo_upc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
X8
X6
)
X6
(
λX7 :
set
⇒
(
¬
atleast5
X5
)
)
)
∧
(
¬
atleast6
X5
)
)
)
∧
(
(
(
(
setsum_p
X4
∧
(
¬
exactly5
(
SNoLev
X5
)
)
)
→
(
¬
atleast5
X6
)
)
→
atleast2
X5
)
∧
(
(
(
(
(
¬
setsum_p
X3
)
∧
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
set_of_pairs
X6
)
)
∧
atleast5
X5
)
∧
(
(
atleast6
X5
→
(
(
(
X5
∈
X6
)
→
(
exactly4
X0
∧
(
¬
exactly2
X6
)
)
→
(
¬
ordinal
X6
)
→
(
¬
exactly2
X0
)
)
∧
(
¬
nat_p
X6
)
)
)
∧
nat_p
X0
)
)
)
)
)
)
)
)
)
)
∧
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
∀X3 :
set
,
∀X4
⊆
X1
,
(
atleast3
X3
∧
(
¬
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
→
(
¬
set_of_pairs
X3
)
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
∀X2
∈
X1
,
(
∃X3 :
set
,
trichotomous_or_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
¬
atleast6
∅
)
)
)
→
(
¬
exactly2
X0
)
)
)
)
)
In Proofgold the corresponding term root is
f16627...
and proposition id is
5510c4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMZdfbQ2pTKeQ9qSq8HYmRvMkCx4PksbPZz
)
∀X0 :
set
,
∀X1 :
set
,
(
(
∃X2 :
set
,
∀X3
⊆
Sing
X0
,
(
(
∃X4 :
set
,
exactly3
X4
)
∧
(
(
¬
atleast3
X0
)
→
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
→
exactly3
X1
)
→
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
In Proofgold the corresponding term root is
e3504e...
and proposition id is
475a16...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMd88FLDAc727XDB64c2rN4XzdLZtaNHejN
)
∃X0 :
set
,
(
(
∃X1 ∈
binintersect
X0
X0
,
∀X2
∈
X1
,
∀X3 :
set
,
(
∀X4 :
set
,
(
(
(
(
¬
exactly5
X2
)
→
atleast3
∅
)
∧
exactly3
(
Sing
X4
)
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast3
X6
)
→
(
¬
atleast6
X4
)
→
(
(
¬
exactly2
X4
)
∧
(
inj
X3
X3
(
λX5 :
set
⇒
X3
)
→
atleast2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
→
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
¬
TransSet
X2
)
)
→
(
(
nat_p
X4
→
(
¬
nat_p
X1
)
)
∧
equip
X4
X2
)
→
(
¬
atleast4
X3
)
)
→
(
∃X4 :
set
,
(
atleast2
X4
∧
exactly3
(
UPair
X3
X2
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
∅
)
∧
(
∀X2 :
set
,
SNo
X2
→
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
(
¬
exactly4
∅
)
∧
(
∀X4 :
set
,
atleast6
(
SNoLev
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0bd526...
and proposition id is
f59d33...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYSn2M9kPj2zpUcv9hk7ckWXNk36n9SAPm
)
∀X0 :
set
,
(
∃X1 :
set
,
(
¬
exactly4
X1
)
→
(
¬
atleast5
X1
)
)
→
(
∃X1 :
set
,
(
atleast6
X1
→
(
∃X2 :
set
,
(
(
∀X3
∈
X1
,
(
¬
atleast4
X1
)
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
→
(
¬
exactly4
X0
)
)
In Proofgold the corresponding term root is
409f2c...
and proposition id is
a26fe2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVwhWcK7LT2ukgF58QYYyG23q2jTU4mxmg
)
∀X0
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
∀X1 :
set
,
(
∀X2 :
set
,
(
∃X3 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
(
SNo
X3
→
(
∃X4 :
set
,
(
¬
ordinal
X4
)
)
)
→
SNo
X2
)
→
(
∀X3
∈
X1
,
∀X4 :
set
,
(
¬
atleast6
X2
)
)
)
→
(
¬
set_of_pairs
X1
)
In Proofgold the corresponding term root is
ee534b...
and proposition id is
4c76ec...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMLPBgTZc9J3zDmjvLGR3MmyYuo6zxz4P37
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3 :
set
,
(
∃X4 :
set
,
(
(
SNo
X4
∧
exactly5
X4
)
∧
TransSet
X4
)
)
→
(
∃X4 :
set
,
(
(
¬
exactly2
X4
)
∧
(
(
(
(
(
(
nat_p
X3
→
(
exactly3
X3
∧
(
(
¬
atleast3
X0
)
∧
(
¬
atleast6
X2
)
)
)
→
(
¬
atleastp
X3
X4
)
)
→
nat_p
X3
)
∧
exactly2
X4
)
∧
atleast2
(
SNoLev
X0
)
)
∧
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
exactly3
X4
∧
(
(
(
(
(
(
exactly5
∅
∧
exactly3
X3
)
→
(
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
exactly3
∅
)
∧
(
¬
atleast3
X2
)
)
∧
(
¬
SNoLe
X3
X4
)
)
→
exactly5
X2
)
)
∧
(
(
(
(
tuple_p
X3
X2
→
(
(
(
(
(
¬
SNo
X3
)
∧
(
(
¬
atleast3
X1
)
→
(
SNoEq_
X1
X3
(
Sing
X1
)
∧
(
¬
ordinal
X3
)
)
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
¬
atleast3
X1
)
)
∧
(
(
exactly2
X0
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
atleast4
X3
)
)
)
∧
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
exactly4
X3
)
)
∧
(
exactly3
X4
∧
(
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
ordinal
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
tuple_p
X6
∅
)
)
∧
(
¬
nat_p
(
ordsucc
X6
)
)
)
)
∧
(
¬
atleast5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
∧
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
exactly3
X1
)
∧
(
(
exactly3
∅
→
(
exactly3
X5
∧
(
atleast2
X5
∧
(
(
(
¬
atleast2
X6
)
→
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
exactly4
X6
)
)
∧
(
(
(
(
(
X6
=
X6
)
→
(
¬
atleast2
X5
)
)
∧
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
∈
X6
)
)
)
∧
TransSet
∅
)
→
(
atleast6
X5
∧
(
exactly5
X5
→
exactly4
X5
)
)
)
)
)
)
)
→
set_of_pairs
X5
)
)
∧
(
¬
setsum_p
X6
)
)
)
)
)
)
)
)
)
)
∧
(
∀X2 :
set
,
(
∀X3
∈
∅
,
(
¬
setsum_p
X0
)
)
→
(
(
∃X3 :
set
,
∃X4 :
set
,
(
atleast4
X0
→
(
¬
exactly5
X2
)
→
(
¬
ordinal
X4
)
)
→
(
atleast4
X3
∧
(
(
¬
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast4
X0
)
)
→
(
atleast3
X3
∧
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
¬
(
X3
∈
X4
)
)
∧
atleast3
X2
)
)
)
→
(
(
(
¬
atleast5
X1
)
→
(
TransSet
X4
→
(
¬
ordinal
X2
)
)
→
atleast5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
setsum_p
X4
)
→
exactly2
X4
)
)
)
∧
(
∃X3 :
set
,
∀X4 :
set
,
(
(
(
(
atleastp
(
ap
X2
∅
)
X4
→
(
¬
ordinal
X2
)
)
∧
(
exactly2
(
𝒫
X0
)
∧
exactly3
X3
)
)
∧
(
¬
(
Inj1
X4
∈
X3
)
)
)
∧
(
(
(
(
(
atleast2
X4
∧
atleast3
X4
)
→
(
(
¬
nat_p
X2
)
∧
exactly4
∅
)
)
→
exactly2
X3
→
TransSet
X2
)
∧
(
¬
atleast4
X2
)
)
→
(
¬
exactly3
X1
)
→
(
¬
SNo
X1
)
→
exactly3
X4
)
)
)
)
)
)
)
→
(
∃X1 :
set
,
(
(
X1
⊆
Inj0
∅
)
∧
(
∀X2
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
∀X3
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
,
∃X4 :
set
,
exactly5
X3
→
(
(
¬
atleast3
X1
)
∧
ordinal
X3
)
→
(
(
¬
TransSet
X3
)
→
(
¬
ordinal
X4
)
)
→
atleast3
∅
)
)
)
In Proofgold the corresponding term root is
ffc6b2...
and proposition id is
643292...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMH1YDdzAiiBvH1vJWR2Nun4ST8C1AequwZ
)
∀X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∃X1 :
set
,
(
(
X1
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
(
(
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
∀X2 :
set
,
(
∀X3 :
set
,
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
X3
)
)
→
(
(
(
(
(
∃X4 ∈
X2
,
(
¬
atleast3
X3
)
)
→
(
∃X4 ∈
X2
,
(
(
¬
atleast2
X1
)
∧
(
(
¬
atleast6
X3
)
→
(
¬
nat_p
X3
)
)
)
)
)
→
(
atleast5
X2
∧
(
(
∀X4 :
set
,
(
¬
setsum_p
X4
)
→
exactly3
X4
)
∧
exactly4
X0
)
)
)
∧
exactly5
(
Inj1
X3
)
)
∧
(
∀X4 :
set
,
(
¬
TransSet
X4
)
→
(
¬
TransSet
X4
)
)
)
)
→
(
(
(
∀X3 :
set
,
∃X4 :
set
,
(
(
¬
exactly2
X4
)
∧
(
¬
nat_p
(
𝒫
X0
)
)
)
)
∧
(
∀X3
∈
X1
,
∀X4
⊆
X0
,
atleast6
X0
)
)
∧
(
∃X3 :
set
,
(
(
X3
⊆
X1
)
∧
(
¬
atleast5
X2
)
)
)
)
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
∧
(
(
¬
atleast4
X1
)
∧
(
∀X2
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
atleast4
X2
)
)
)
)
In Proofgold the corresponding term root is
ee5783...
and proposition id is
50bc8a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMbDpn2KqvkErnCHriG4C1R7P27tB6cNxuK
)
∀X0
∈
∅
,
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
∀X2 :
set
,
(
∀X3
⊆
X2
,
∀X4
⊆
⋃
X3
,
(
(
atleast5
∅
∧
(
¬
exactly4
X3
)
)
∧
(
exactly2
X3
∧
(
¬
ordinal
X4
)
)
)
→
(
¬
ordinal
(
proj1
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
set_of_pairs
(
In_rec_i
(
λX3 :
set
⇒
λX4 :
set
→
set
⇒
X3
)
X2
)
)
)
In Proofgold the corresponding term root is
37bbc6...
and proposition id is
fa1d0e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMdsurnwKugxPJjtWxjRZZncG5mw6QF5t8H
)
∃X0 :
set
,
(
(
∀X1
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
∀X4 :
set
,
(
(
¬
atleast3
X2
)
→
ordinal
X2
)
→
(
(
atleast4
X3
∧
(
(
atleast6
X3
∧
(
¬
exactly3
X3
)
)
∧
(
¬
atleast3
X4
)
)
)
∧
(
¬
setsum_p
∅
)
)
→
SNoLt
X4
∅
)
)
)
∧
(
∃X3 :
set
,
atleast2
X2
)
)
)
∧
(
∀X1
⊆
X0
,
(
X0
∈
X0
)
)
)
In Proofgold the corresponding term root is
ddf77a...
and proposition id is
e678bc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMGu8sTcncXK8zBCCWcM7dusc8au9PBu8T
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
exactly4
X1
∧
(
∀X2 :
set
,
(
¬
atleast4
X1
)
→
(
(
(
∀X3
∈
X1
,
(
(
¬
exactly4
X1
)
∧
(
(
(
∃X4 :
set
,
binop_on
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X2
)
)
∧
(
∃X4 :
set
,
(
(
(
(
(
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
atleast6
X1
∧
(
atleast2
X3
∧
(
¬
atleast6
X4
)
)
)
∧
(
¬
nat_p
X4
)
)
)
∧
(
¬
TransSet
X0
)
)
→
(
¬
TransSet
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
(
(
atleast2
X4
→
(
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
set_of_pairs
X0
∧
exactly4
X4
)
)
)
→
(
(
exactly2
X4
∧
(
¬
exactly5
X3
)
)
∧
(
¬
ordinal
X4
)
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
(
¬
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
ordinal
X1
)
)
)
)
∧
atleast3
X4
)
)
)
∧
(
¬
nat_p
X2
)
)
)
)
∧
(
∀X3 :
set
,
atleast2
X3
)
)
∧
(
(
(
∀X3
∈
X2
,
(
(
(
∃X4 :
set
,
(
exactly5
X4
∧
(
(
(
(
⋃
X2
∈
X1
)
∧
(
atleast5
X4
→
nat_p
X3
)
)
∧
(
(
¬
exactly3
X2
)
∧
(
(
atleast3
X2
∧
(
(
(
(
¬
atleast5
X2
)
→
atleast5
∅
)
∧
(
atleast6
X3
→
atleast2
X4
→
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
exactly3
X4
)
)
∧
(
¬
exactly3
X0
)
)
)
)
→
(
¬
atleast4
X4
)
)
)
)
→
setsum_p
∅
)
∧
(
∃X4 :
set
,
(
(
(
¬
exactly3
X1
)
→
(
¬
exactly2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
exactly2
X2
)
)
)
)
→
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
(
∃X4 :
set
,
(
¬
SNo
X2
)
)
→
(
¬
atleast5
X1
)
)
)
)
)
∧
(
¬
atleast6
X0
)
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
(
(
∃X2 :
set
,
(
(
∀X3
⊆
X2
,
(
¬
transitive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
(
(
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
exactly2
X5
→
(
¬
set_of_pairs
X1
)
)
)
→
nat_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
PNoLt_
X4
(
λX6 :
set
⇒
(
¬
atleast3
X0
)
)
(
λX6 :
set
⇒
(
exactly3
X6
∧
(
exactly5
X6
∧
atleast5
X5
)
)
)
)
∧
(
(
¬
ordinal
X1
)
∧
(
exactly2
X3
∧
(
¬
exactly2
X1
)
)
)
)
)
)
)
∧
(
∃X3 :
set
,
(
(
∀X4 :
set
,
(
(
(
(
exactly2
X4
→
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
exactly3
X3
∧
(
¬
exactly5
X3
)
)
)
)
∧
(
atleast6
X4
∧
(
¬
atleast3
X3
)
)
)
∧
(
(
(
TransSet
X1
∧
(
¬
exactly3
X2
)
)
∧
(
(
¬
atleast3
X4
)
→
(
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
TransSet
X3
)
)
∧
(
¬
SNo_
(
ReplSep
∅
(
λX5 :
set
⇒
atleast2
X1
)
(
λX5 :
set
⇒
∅
)
)
X3
)
)
)
)
∧
(
atleast2
X3
∧
(
¬
atleast4
∅
)
)
)
)
)
∧
(
(
atleast2
∅
→
(
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
(
(
¬
exactly4
(
V_
X4
)
)
∧
(
(
¬
atleast5
∅
)
→
atleast4
X4
)
)
)
)
)
∧
(
∀X4
⊆
X2
,
(
¬
setsum_p
X0
)
)
)
)
)
)
)
→
(
∀X2 :
set
,
(
∃X3 :
set
,
(
(
X3
⊆
X1
)
∧
atleast3
X3
)
)
→
exactly3
∅
→
(
(
setsum_p
X2
→
(
∀X3
⊆
X1
,
∃X4 ∈
V_
∅
,
atleast2
X3
→
(
(
¬
exactly4
X4
)
∧
(
¬
nat_p
X3
)
)
→
(
atleast4
(
famunion
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
X4
)
)
∧
(
exactly4
X4
→
exactly2
X3
)
)
)
)
∧
exactly4
(
V_
X2
)
)
)
)
→
SNo
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
In Proofgold the corresponding term root is
83b861...
and proposition id is
29e24d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcrvt4EaWx8p9AuGo2ygyj9LQ6dBZKxuev
)
∀X0 :
set
,
(
∀X1
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
atleast2
X1
→
TransSet
∅
)
→
(
∃X1 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
,
∃X2 ∈
X1
,
(
∃X3 :
set
,
(
(
X3
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
∀X4 :
set
,
(
atleast5
X0
→
(
¬
atleast4
X4
)
)
→
setsum_p
X2
)
)
)
→
(
∀X3
⊆
X1
,
(
(
(
∀X4 :
set
,
(
SNoLe
X1
X4
∧
(
¬
exactly3
X3
)
)
→
(
(
(
¬
tuple_p
X3
X3
)
→
(
setsum_p
(
binunion
X4
X4
)
∧
TransSet
X1
)
)
∧
PNoLt_
(
⋃
X3
)
(
λX5 :
set
⇒
atleast3
X4
)
(
λX5 :
set
⇒
(
¬
nat_p
X4
)
)
)
)
→
(
∀X4
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
,
(
¬
atleast6
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
∃X4 :
set
,
(
(
(
(
¬
TransSet
X4
)
→
(
(
(
¬
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
ordinal
X5
)
)
)
∧
SNoEq_
X3
X4
X3
)
∧
(
SNo
∅
∧
exactly2
X4
)
)
)
∧
(
¬
setsum_p
X3
)
)
∧
(
nat_p
X0
→
(
atleast4
X4
∧
(
¬
atleast2
X3
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
efcb19...
and proposition id is
1eaa59...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMaFHYVknhemg7Mm9KBpfez1FT5JdoxErcJ
)
∃X0 :
set
,
(
(
X0
⊆
Sep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
λX1 :
set
⇒
∃X2 :
set
,
(
(
X2
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
(
¬
set_of_pairs
X2
)
)
)
)
∧
(
∀X1 :
set
,
(
∀X2
∈
X0
,
(
¬
atleast6
X1
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3 :
set
,
∀X4 :
set
,
(
(
(
(
X2
=
X4
)
→
(
¬
atleast5
(
binintersect
X1
X4
)
)
)
∧
(
(
(
¬
ordinal
X2
)
→
(
(
¬
atleast2
X2
)
∧
(
atleast5
X2
∧
(
(
(
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
atleast6
X3
)
)
→
(
¬
exactly4
X3
)
)
∧
nat_p
X2
)
)
)
)
∧
(
¬
SNo
X3
)
)
)
∧
(
¬
nat_p
X2
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fd875c...
and proposition id is
d63bc7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYDRbxo33KRmxxPBmp4Ee3u9uUm7DLyVhb
)
∀X0
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
,
∃X1 ∈
X0
,
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
¬
exactly3
(
lam
X1
(
λX3 :
set
⇒
X3
)
)
)
→
(
∃X3 :
set
,
(
(
X3
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
∃X4 :
set
,
(
(
¬
atleast6
X4
)
∧
exactly2
X4
)
)
→
atleast4
X3
)
)
)
)
)
In Proofgold the corresponding term root is
e9a2e2...
and proposition id is
ec3c29...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMGwevTsYLn6zUM8wFMqL4bA7eab3Pfd5QS
)
∀X0
∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
∀X1
⊆
X0
,
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
(
∀X3 :
set
,
(
(
∃X4 :
set
,
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
nat_p
X4
)
∧
(
∃X4 :
set
,
(
¬
exactly5
X1
)
)
)
→
(
∀X4 :
set
,
nat_p
X3
)
)
∧
(
(
¬
atleast3
X2
)
∧
exactly4
X2
)
)
)
)
→
SNoLt
X1
X0
In Proofgold the corresponding term root is
424fbc...
and proposition id is
0b9f36...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMPTJg4z5s8JpcNVzqCsjVh5wCzt7kpfRa4
)
∃X0 :
set
,
(
(
∀X1 :
set
,
(
(
∀X2
∈
X0
,
∃X3 :
set
,
(
irreflexive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
¬
tuple_p
X5
X5
)
)
∧
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
∀X2 :
set
,
(
(
∃X3 :
set
,
(
(
X3
⊆
X1
)
∧
(
(
(
∃X4 ∈
X3
,
(
(
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
(
(
atleast2
X3
→
(
TransSet
X3
∧
(
(
(
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
(
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
exactly4
X4
∧
atleastp
X1
X1
)
)
→
(
¬
ordinal
X3
)
)
)
→
(
¬
(
X3
∈
X4
)
)
)
→
(
¬
SNo_
X3
X3
)
)
)
)
∧
exactly5
X4
)
)
∧
setsum_p
X1
)
)
∧
(
(
¬
exactly2
X0
)
→
(
∃X4 :
set
,
(
¬
setsum_p
X4
)
)
→
exactly5
X2
)
)
∧
(
∀X4
⊆
X1
,
(
¬
atleast3
X0
)
→
(
¬
exactly5
(
SNoLev
X3
)
)
)
)
)
)
∧
(
∃X3 :
set
,
(
(
∀X4 :
set
,
(
(
(
(
atleast3
X1
→
(
(
(
¬
SNoLe
∅
X1
)
→
(
¬
exactly2
X4
)
)
∧
exactly5
∅
)
)
→
nat_p
∅
)
∧
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
(
(
atleast4
X1
∧
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
SNo
X0
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
(
atleast5
X3
→
(
¬
(
X1
∈
X2
)
)
)
∧
(
¬
(
X2
∈
X0
)
)
)
∧
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
∈
X3
)
)
)
∧
(
(
atleast4
X3
∧
(
¬
ordinal
∅
)
)
→
(
∃X4 ∈
X0
,
(
¬
atleast4
X0
)
→
(
¬
nat_p
X3
)
)
)
)
)
)
→
(
∀X3 :
set
,
(
∀X4 :
set
,
(
setsum_p
X2
∧
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
(
(
∀X4
⊆
X1
,
(
atleast2
X2
→
(
(
set_of_pairs
∅
∧
(
(
(
(
¬
TransSet
X0
)
→
(
¬
atleast3
X4
)
→
atleast3
X4
)
→
(
(
¬
exactly5
∅
)
∧
(
(
¬
exactly2
X2
)
→
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
ordinal
X2
)
)
∧
ordinal
(
Pi
X3
(
λX5 :
set
⇒
X4
)
)
)
)
→
(
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
TransSet
(
⋃
X3
)
)
)
)
∧
(
∀X4 :
set
,
atleast3
(
nat_primrec
X0
(
λX5 :
set
⇒
λX6 :
set
⇒
X5
)
X4
)
→
(
¬
ordinal
X3
)
)
)
∧
(
∀X4
∈
X1
,
(
¬
SNoLe
X2
X3
)
→
(
(
(
¬
atleast6
X3
)
→
(
(
¬
atleast3
X3
)
∧
exactly4
X3
)
)
→
set_of_pairs
X0
)
→
(
¬
atleast5
X4
)
)
)
)
)
)
)
∧
(
∀X1 :
set
,
∀X2
⊆
X0
,
exactly5
X1
)
)
In Proofgold the corresponding term root is
bb12f8...
and proposition id is
80257d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMKiEVp78PBwxV5kBBRkHR3qw4ZnAoJtG6G
)
∃X0 :
set
,
(
(
X0
⊆
∅
)
∧
(
∃X1 :
set
,
(
(
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
∃X3 ∈
X2
,
∀X4
⊆
X2
,
(
(
(
(
(
(
exactly4
∅
→
atleast3
X4
)
∧
(
¬
set_of_pairs
X4
)
)
∧
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
ordinal
∅
)
∧
(
(
¬
setsum_p
∅
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
¬
atleast2
∅
)
)
→
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
∀X2
∈
X1
,
∃X3 :
set
,
∃X4 :
set
,
(
(
¬
setsum_p
(
SNoLev
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
¬
SNoLt
X4
X4
)
∧
(
¬
SNo
(
Sing
X2
)
)
)
→
(
exactly4
∅
→
(
(
(
¬
exactly3
X4
)
→
atleast3
X1
→
(
¬
SNo
X4
)
→
(
exactly4
X4
∧
(
¬
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
ordinal
X5
)
)
)
)
∧
(
(
exactly2
X2
∧
(
¬
atleast3
X4
)
)
∧
(
PNo_downc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
X6
X4
∧
(
(
(
¬
X6
X5
)
∧
(
(
(
(
(
¬
X6
X1
)
∧
(
¬
(
X2
⊆
X5
)
)
)
→
(
¬
ordinal
X0
)
)
→
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
SNo_
X0
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
atleast6
X5
)
∧
X6
∅
)
)
→
(
¬
X6
X4
)
)
∧
(
(
¬
X6
X5
)
∧
(
X6
X5
→
exactly3
X0
)
)
)
)
∧
(
¬
exactly5
X3
)
)
)
)
X3
(
λX5 :
set
⇒
(
(
(
¬
atleast5
∅
)
→
(
¬
PNoLt_
X5
(
λX6 :
set
⇒
(
¬
atleast5
X6
)
)
(
λX6 :
set
⇒
TransSet
X6
)
)
)
∧
exactly3
X2
)
)
∧
(
(
(
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast2
X4
)
→
(
(
(
¬
atleast6
X4
)
→
(
¬
(
ordsucc
X4
⊆
X1
)
)
)
∧
(
(
SNo_
X3
X0
∧
exactly2
X1
)
→
(
TransSet
X0
∧
(
¬
equip
X4
X4
)
)
)
)
→
exactly5
X2
)
→
(
(
exactly3
X0
→
atleast6
X4
)
→
atleast4
X4
)
→
(
atleast4
X4
∧
(
¬
(
X2
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
¬
atleast2
X2
)
)
)
)
)
→
(
¬
TransSet
∅
)
)
→
(
(
(
(
(
¬
exactly1of2
(
atleast2
X2
)
(
(
(
(
X4
=
X3
)
∧
atleast5
X3
)
∧
(
¬
atleast5
∅
)
)
∧
TransSet
X3
)
)
→
(
exactly5
X2
∧
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X3
)
)
)
)
∧
(
(
(
(
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
ordinal
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
¬
exactly2
∅
)
→
atleast5
∅
)
)
→
atleast4
X1
→
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
nat_p
X3
)
∧
(
(
(
(
atleast6
X0
→
SNo
X3
)
∧
(
¬
atleast3
X4
)
)
→
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
exactly3
X3
)
)
)
∧
exactly3
X3
)
∧
(
¬
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast5
X6
→
(
ordinal
X6
∧
(
(
(
(
setsum_p
X2
→
TransSet
X6
→
atleast6
X6
)
∧
(
(
exactly4
∅
∧
(
(
(
(
atleast2
X1
∧
(
nat_p
X6
→
(
(
(
exactly5
X6
→
(
(
¬
exactly4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
atleast6
X6
)
)
)
→
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast4
X8
)
)
)
)
∧
(
(
atleast6
X5
→
(
(
¬
atleast3
X5
)
∧
nat_p
X5
)
)
∧
(
(
(
¬
exactly3
X0
)
→
(
(
¬
nat_p
X5
)
∧
(
(
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
(
exactly4
(
Sing
∅
)
→
(
¬
ordinal
(
PSNo
X5
(
λX7 :
set
⇒
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
exactly4
X9
)
)
)
→
exactly4
X0
)
)
)
→
exactly2
X5
)
→
(
¬
nat_p
X5
)
)
→
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
¬
exactly3
X6
)
)
)
)
)
)
→
(
nat_p
X6
∧
(
¬
(
X5
∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
exactly5
X5
)
→
exactly5
X6
)
)
→
totalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
TransSet
X7
)
→
nat_p
X8
)
→
exactly2
X6
)
)
)
∧
exactly3
(
binunion
X6
X6
)
)
→
(
(
¬
antisymmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
set_of_pairs
X3
)
)
)
∧
(
(
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
exactly4
X6
)
)
→
atleast4
X2
)
)
)
)
)
)
)
∧
(
(
(
(
¬
SNo
X0
)
∧
(
(
SNoLe
(
proj0
X4
)
∅
∧
(
exactly5
X3
→
(
¬
atleast5
X2
)
)
)
→
(
(
¬
setsum_p
X3
)
∧
(
(
¬
exactly2
X0
)
→
(
(
SNo
X1
→
(
(
(
¬
SNo_
X4
X3
)
→
(
atleastp
X4
(
Sing
X4
)
→
ordinal
X1
)
→
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
atleast2
X4
∧
(
¬
ordinal
X4
)
)
)
∧
(
SNo
X2
→
(
¬
TransSet
X4
)
)
)
→
ordinal
X4
)
→
ordinal
X3
)
→
(
¬
exactly5
X3
)
)
)
)
)
∧
(
exactly2
X2
∧
atleast4
X3
)
)
→
exactly2
X4
)
)
)
→
SNo
(
V_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
¬
atleast4
X2
)
∧
(
exactly5
X4
→
TransSet
X4
)
)
)
)
→
(
(
(
(
(
(
(
set_of_pairs
X1
∧
(
¬
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
→
(
¬
PNoLe
X1
(
λX5 :
set
⇒
(
SNo
X4
→
set_of_pairs
X3
)
→
ordinal
X5
)
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
exactly4
X6
)
)
∧
(
¬
exactly4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
→
(
¬
atleast5
X1
)
→
(
atleast2
X2
∧
atleast6
X2
)
)
∧
exactly2
∅
)
∧
(
(
¬
(
X2
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
exactly2
X1
)
→
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
¬
atleast5
X2
)
)
∧
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
¬
exactly5
X0
)
)
)
∧
(
∀X2 :
set
,
∀X3 :
set
,
∀X4
⊆
X2
,
(
(
(
(
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
(
¬
atleast4
X4
)
∧
atleast4
X2
)
→
atleast6
X0
)
)
∧
(
(
¬
exactly3
X3
)
∧
(
(
¬
atleast6
X1
)
∧
(
(
¬
atleast4
X3
)
→
(
(
(
¬
SNo
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
SNo_
X3
X3
∧
(
¬
nat_p
X4
)
)
∧
(
(
(
(
¬
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
nat_p
X5
)
)
∧
(
¬
ordinal
X4
)
)
→
(
(
(
(
(
exactly2
(
Inj1
X1
)
∧
(
(
(
atleast3
X3
∧
setsum_p
X4
)
∧
(
¬
nat_p
X4
)
)
∧
(
(
¬
exactly2
X3
)
∧
(
(
(
¬
exactly3
X3
)
→
(
¬
exactly5
X4
)
)
→
(
(
¬
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
¬
exactly5
X4
)
)
)
)
)
)
∧
(
¬
ordinal
X0
)
)
→
nat_p
X3
)
∧
(
¬
nat_p
X0
)
)
∧
(
(
(
(
(
(
¬
atleast6
X3
)
→
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast3
X3
)
)
∧
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
¬
SNo
X3
)
→
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast2
X0
)
)
)
)
→
(
¬
PNoLe
X3
(
λX5 :
set
⇒
ordinal
X4
)
X0
(
λX5 :
set
⇒
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
)
)
∧
(
(
(
atleast3
X4
→
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
(
(
(
(
(
(
¬
nat_p
X1
)
→
(
X0
∈
∅
)
→
(
(
¬
nat_p
X2
)
∧
(
¬
exactly5
X3
)
)
)
→
(
¬
exactly5
X2
)
→
set_of_pairs
(
⋃
X1
)
)
→
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
(
atleast2
X4
→
exactly5
X6
→
(
¬
exactly2
X5
)
)
∧
(
(
¬
ordinal
X6
)
→
(
(
(
linear_i
(
λX7 :
set
⇒
λX8 :
set
⇒
exactly5
X8
)
∧
(
symmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
ordinal
X8
)
→
(
¬
ordinal
X6
)
)
)
∧
exactly2
X6
)
∧
exactly3
X6
)
)
)
→
equip
X5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
(
(
¬
strictpartialorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
SNoLt
X7
∅
)
∧
(
ordinal
X7
→
atleast3
X7
)
)
)
)
∧
(
(
¬
TransSet
X3
)
→
exactly5
X6
)
)
∧
(
(
(
¬
atleastp
X5
X5
)
→
(
(
(
¬
TransSet
∅
)
∧
(
(
(
ordinal
X1
→
(
atleast6
X0
∧
(
¬
SNo_
X5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
¬
set_of_pairs
X0
)
→
exactly3
X5
)
)
∧
(
(
(
(
(
atleast6
X5
∧
(
¬
PNoEq_
X0
(
λX7 :
set
⇒
(
(
(
¬
TransSet
X7
)
→
(
¬
TransSet
X4
)
→
(
(
exactly5
∅
∧
(
TransSet
X0
→
atleast2
X6
)
)
∧
(
(
(
atleast3
X7
∧
(
¬
setsum_p
X6
)
)
→
(
¬
ordinal
X6
)
→
atleast3
X1
)
→
atleast5
∅
)
)
)
→
(
exactly4
X7
∧
(
(
(
exactly3
∅
→
ordinal
X1
)
→
(
¬
atleast6
X7
)
)
∧
(
¬
exactly2
X6
)
)
)
)
→
TransSet
X6
)
(
λX7 :
set
⇒
(
¬
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
→
exactly4
X5
)
→
(
¬
exactly3
X6
)
)
→
tuple_p
X6
X6
→
(
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
exactly5
(
𝒫
X4
)
)
→
(
¬
ordinal
X0
)
→
exactly2
X6
)
→
(
(
(
(
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
exactly2
X5
)
∧
(
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
exactly4
X8
)
∧
(
(
¬
TransSet
X8
)
∧
(
exactly4
X8
→
(
¬
transitive_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
ordinal
X4
∧
(
TransSet
X9
∧
(
TransSet
X9
→
(
¬
(
X10
∈
X9
)
)
)
)
)
)
)
)
)
)
)
→
(
(
exactly4
X6
∧
ordinal
(
binrep
X6
X5
)
)
∧
SNo
X6
)
)
∧
(
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
→
(
¬
atleastp
X4
X5
)
)
→
(
(
¬
exactly2
X0
)
∧
(
(
¬
atleast5
X5
)
→
exactly4
X6
)
)
)
∧
(
(
¬
TransSet
X5
)
∧
(
(
(
(
(
atleast3
X0
→
TransSet
X3
)
→
(
(
¬
exactly3
X5
)
∧
(
(
¬
ordinal
X0
)
→
atleast4
X0
→
setsum_p
X6
→
(
(
¬
atleast2
X2
)
∧
exactly2
∅
)
)
)
)
→
atleast3
X6
→
TransSet
(
UPair
X5
X5
)
)
→
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
atleast5
X4
→
(
(
¬
atleast4
X6
)
∧
atleast4
X6
)
)
∧
(
(
atleast5
X5
∧
exactly3
X6
)
∧
(
(
¬
setsum_p
X5
)
→
(
¬
atleast4
X6
)
)
)
)
∧
(
¬
exactly3
X5
)
)
)
)
)
)
→
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
exactly2
∅
→
TransSet
X5
)
)
)
)
→
(
(
(
nat_p
X5
→
(
(
¬
exactly5
X5
)
∧
ordinal
X0
)
)
→
(
(
¬
exactly2
∅
)
→
nat_p
X0
→
(
(
¬
atleast4
X6
)
→
set_of_pairs
X5
)
→
(
¬
exactly3
X5
)
)
→
(
¬
exactly4
(
Unj
(
𝒫
X5
)
)
)
)
∧
(
(
¬
atleast3
X5
)
→
(
(
atleast4
X6
∧
(
¬
atleast2
X4
)
)
∧
SNo_
X5
(
𝒫
X4
)
)
)
)
)
)
∧
(
¬
exactly2
(
ordsucc
(
Inj0
X4
)
)
)
)
→
(
¬
TransSet
X3
)
)
∧
(
(
¬
atleast3
X1
)
→
(
(
(
(
¬
atleast4
X4
)
∧
exactly4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
(
SNo
X4
∧
(
¬
ordinal
X0
)
)
∧
atleast3
X3
)
∧
(
nat_p
X4
∧
(
nat_p
(
ordsucc
X2
)
→
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
→
(
¬
atleast2
X4
)
)
∧
(
¬
equip
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
X4
)
)
)
)
∧
(
(
(
ordinal
∅
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
nat_p
X5
∧
(
PNoLe
X2
(
λX7 :
set
⇒
TransSet
X6
→
(
(
(
(
nat_p
X7
→
atleast4
(
UPair
X2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
¬
atleast5
X0
)
∧
(
exactly5
X0
→
(
¬
nat_p
X7
)
)
)
)
→
(
atleast2
X7
∧
(
set_of_pairs
X6
→
(
¬
atleast6
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
→
setsum_p
X4
)
→
atleast2
X7
)
→
(
exactly5
∅
∧
(
X2
∈
X6
)
)
)
X6
(
λX7 :
set
⇒
(
¬
SNo
∅
)
→
atleast6
X7
)
→
(
(
¬
atleast4
X6
)
→
(
¬
nat_p
X6
)
)
→
(
¬
(
X6
∈
X6
)
)
)
)
)
)
→
(
(
¬
atleast3
X4
)
∧
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
¬
exactly3
∅
)
)
→
tuple_p
X3
X0
)
)
)
→
(
setsum_p
X3
∧
atleast6
X3
)
→
exactly3
X1
→
(
¬
set_of_pairs
X3
)
)
)
)
)
)
)
∧
(
atleast6
X2
→
(
¬
nat_p
X4
)
)
)
∧
(
atleastp
X4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∧
(
atleast3
X4
∧
exactly5
X1
)
)
)
∧
(
¬
exactly3
X2
)
)
)
)
)
)
In Proofgold the corresponding term root is
5b2617...
and proposition id is
ee16d3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVixMcejXURDNZndYwXpogcwCpVfq4B7FG
)
∃X0 :
set
,
(
(
∃X1 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∃X2 :
set
,
(
(
X2
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
¬
atleast2
X2
)
∧
(
¬
nat_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
TransSet
X1
∧
(
¬
exactly2
X1
)
)
∧
(
∃X2 :
set
,
(
∀X3 :
set
,
(
(
¬
equip
X3
∅
)
→
(
exactly2
X3
∧
exactly4
X2
)
)
→
(
(
∀X4 :
set
,
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
∀X4 :
set
,
atleast3
X4
→
(
(
(
(
PNoLe
X4
(
λX5 :
set
⇒
(
(
(
(
PNoLt
X2
(
λX6 :
set
⇒
exactly2
X6
)
X4
(
λX6 :
set
⇒
(
(
¬
atleast4
(
V_
X5
)
)
∧
(
(
(
¬
atleast5
X5
)
∧
exactly5
X5
)
→
exactly3
X0
→
(
(
exactly3
X6
∧
exactly2
X5
)
∧
(
(
¬
SNo
X6
)
∧
(
(
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
(
(
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
(
TransSet
X5
∧
(
¬
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
=
X6
)
)
)
)
∧
(
(
(
¬
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast3
X4
)
→
(
¬
atleast5
X5
)
)
)
→
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleastp
X8
X7
)
)
)
∧
(
¬
atleast6
X2
)
)
)
)
)
)
→
(
atleastp
(
Sing
X6
)
X3
∧
(
exactly5
∅
∧
(
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast5
(
SNoLev
∅
)
)
)
)
)
)
∧
(
(
(
¬
atleast3
∅
)
∧
(
¬
SNo
X2
)
)
∧
(
(
¬
atleast3
X5
)
∧
(
(
(
(
(
(
atleast4
(
If_i
(
(
¬
exactly5
X2
)
∧
(
¬
atleast3
X4
)
)
∅
X4
)
∧
atleast5
X4
)
→
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
atleast2
X5
→
(
¬
atleast3
X5
)
)
∧
(
(
PNoLt
X5
(
λX6 :
set
⇒
equip
X2
X2
→
(
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
¬
exactly3
X3
)
)
→
(
TransSet
X5
∧
SNo
X5
)
)
→
(
atleast2
X6
∧
(
(
¬
exactly5
X1
)
→
(
¬
PNo_upc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
(
(
(
¬
X8
X4
)
∧
(
X8
X6
∧
ordinal
X6
)
)
∧
(
(
(
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
⊆
X0
)
∧
(
¬
exactly3
(
Inj0
X7
)
)
)
)
∧
(
(
X8
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
exactly5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
¬
atleast5
X7
)
)
)
→
(
exactly4
(
V_
∅
)
∧
per_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
¬
atleast4
X4
)
)
)
)
(
nat_primrec
(
Unj
X0
)
(
λX7 :
set
⇒
λX8 :
set
⇒
X2
)
X0
)
(
λX7 :
set
⇒
(
(
¬
exactly2
X7
)
∧
nat_p
X6
)
→
(
equip
X2
∅
∧
TransSet
X7
)
)
)
)
)
)
X4
(
λX6 :
set
⇒
transitive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
exactly4
X8
)
)
)
∧
(
(
¬
TransSet
X4
)
→
(
¬
set_of_pairs
X5
)
)
)
∧
exactly3
X0
)
)
∧
(
(
(
¬
ordinal
X0
)
→
(
(
(
(
(
(
(
(
(
¬
SNo
X4
)
∧
(
¬
atleast5
X4
)
)
∧
(
¬
atleastp
X2
X1
)
)
→
(
¬
atleast5
X4
)
)
→
(
(
SNo_
X4
X5
→
(
¬
exactly3
X5
)
)
→
(
¬
atleast2
X2
)
)
→
(
¬
exactly5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
ordinal
X5
)
∧
(
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
nat_p
X0
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
¬
atleastp
X1
X4
)
)
∧
(
(
¬
nat_p
(
⋃
X4
)
)
→
(
atleast4
∅
∧
(
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
∈
X0
)
∧
(
set_of_pairs
X5
→
(
atleast2
X5
∧
(
¬
atleast3
(
ap
(
ordsucc
X4
)
X4
)
)
)
)
)
)
)
)
)
→
(
atleast2
X4
∧
(
ordinal
X1
∧
(
(
atleast5
X0
→
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
¬
exactly4
∅
)
→
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
∧
(
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
atleast4
X0
)
)
→
exactly2
(
Unj
X4
)
)
∧
exactly2
X4
)
)
∧
(
(
¬
atleast2
X4
)
∧
(
(
atleast5
X5
→
(
¬
atleast4
X5
)
)
∧
atleast4
X5
)
)
)
)
)
)
→
(
(
¬
exactly3
X3
)
∧
atleast6
X4
)
)
→
atleast6
X4
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
X4
(
λX5 :
set
⇒
exactly5
X3
→
(
¬
TransSet
X5
)
→
(
¬
exactly5
∅
)
→
(
¬
atleast6
X4
)
)
→
TransSet
X3
)
→
(
(
(
(
(
atleast4
X2
→
(
exactly2
X1
→
(
¬
exactly2
X4
)
)
→
(
TransSet
X3
∧
SNoEq_
X4
(
SetAdjoin
X4
X2
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
¬
exactly3
X1
)
∧
(
(
(
(
¬
atleast5
X1
)
→
(
¬
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast4
(
Inj0
X6
)
)
→
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
→
(
(
¬
exactly5
(
ap
X3
∅
)
)
∧
(
(
(
¬
equip
X2
X0
)
→
(
(
(
¬
exactly4
X4
)
→
(
atleast3
X2
∧
(
exactly3
X0
∧
(
¬
exactly4
X4
)
)
)
)
→
atleast2
X3
→
ordinal
X2
)
→
(
(
¬
atleast6
X3
)
→
atleast4
X1
)
→
atleast2
X3
)
∧
tuple_p
X4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
(
ordinal
X1
∧
(
exactly3
X3
→
(
X1
∈
X4
)
)
)
∧
ordinal
X2
)
)
→
(
¬
TransSet
X4
)
)
)
)
∧
(
(
(
(
setsum_p
X3
→
(
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast5
X3
)
)
)
∧
atleast2
X2
)
→
SNo
X2
→
(
¬
nat_p
X2
)
)
→
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
TransSet
X4
)
∧
(
¬
exactly3
X4
)
)
)
∧
(
(
(
(
(
¬
atleast6
X3
)
∧
(
(
¬
atleast4
X2
)
∧
(
atleast4
X4
→
(
¬
exactly2
(
V_
X0
)
)
)
)
)
∧
(
(
atleast4
X0
→
(
¬
exactly2
X4
)
)
∧
(
¬
nat_p
(
⋃
X3
)
)
)
)
→
(
atleast2
X4
→
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast3
X6
)
)
∧
atleast5
X2
)
∧
(
¬
atleast5
X3
)
)
)
→
(
(
(
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
atleast5
X3
)
→
(
atleast6
X2
∧
(
¬
exactly4
X4
)
)
)
∧
(
ordinal
X4
∧
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly3
X5
)
)
)
)
∧
(
(
¬
atleast5
X3
)
∧
(
¬
TransSet
X3
)
)
)
)
∧
exactly3
X3
)
)
)
)
→
(
∃X3 :
set
,
(
(
X3
⊆
∅
)
∧
exactly3
X1
)
)
)
)
)
)
In Proofgold the corresponding term root is
3b7eb0...
and proposition id is
177a9c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMnUesxvyUL6bRzWvP9pjfWuYUkaxsFnG4
)
∃X0 :
set
,
∀X1
⊆
X0
,
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
¬
(
X2
⊆
X2
)
)
→
(
∃X3 :
set
,
(
atleast6
(
binunion
X1
X2
)
∧
(
∀X4 :
set
,
(
¬
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast5
X5
)
→
atleast3
X0
)
)
→
ordinal
X2
)
)
)
)
)
)
→
(
∃X2 :
set
,
TransSet
X0
)
In Proofgold the corresponding term root is
ce249e...
and proposition id is
98f810...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMJ6ZeDUw1aJhA4CfoNYh8F9HCaP5mSdgeF
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
(
∀X2
⊆
X0
,
(
(
¬
nat_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
reflexive_i
(
λX3 :
set
⇒
λX4 :
set
⇒
atleast2
X2
)
)
)
)
∧
(
¬
atleast4
X0
)
)
)
∧
(
∃X1 :
set
,
(
¬
setsum_p
X1
)
→
(
(
¬
atleastp
X0
X1
)
∧
(
∃X2 :
set
,
(
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
∃X3 :
set
,
(
(
(
¬
exactly4
X0
)
→
(
∃X4 :
set
,
(
(
(
atleast6
X2
∧
(
¬
atleast4
(
𝒫
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
ordinal
X4
)
∧
(
(
(
¬
atleast5
X3
)
→
(
(
¬
atleast5
X2
)
∧
(
(
(
(
exactly5
X0
→
(
(
¬
atleast2
X4
)
∧
(
¬
atleast4
X4
)
)
)
∧
(
¬
exactly4
X4
)
)
→
ordinal
X4
)
→
(
¬
(
X3
∈
X3
)
)
)
)
→
(
¬
exactly3
X3
)
)
∧
(
¬
TransSet
X4
)
)
)
)
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
(
¬
exactly5
X2
)
∧
(
(
¬
exactly4
X2
)
→
(
(
(
(
¬
TransSet
X4
)
∧
ordinal
X4
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast4
X5
)
)
)
∧
(
(
ordinal
X3
∧
exactly2
X1
)
→
(
¬
(
𝒫
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
=
X3
)
)
)
)
)
)
→
atleast4
X4
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
20f60f...
and proposition id is
807835...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMbYhqVfLkEJK36z7kmbrVppsmkprngBiAs
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
∀X1 :
set
,
(
(
(
∃X2 :
set
,
(
atleast5
X2
∧
(
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
∃X4 :
set
,
(
¬
nat_p
X0
)
)
)
)
)
)
∧
(
∀X2 :
set
,
∃X3 :
set
,
∃X4 :
set
,
(
¬
atleast6
X4
)
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
exactly3
X0
)
)
→
(
∀X2 :
set
,
(
∀X3
⊆
SetAdjoin
(
add_nat
X1
X1
)
X0
,
atleast2
X2
)
→
(
(
(
∃X3 :
set
,
(
(
(
∃X4 ∈
X0
,
(
(
(
¬
TransSet
X2
)
∧
exactly4
X1
)
∧
(
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
atleast3
X5
→
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast6
X8
→
(
(
(
(
¬
ordinal
(
SNoElts_
X4
)
)
→
(
¬
atleast3
X5
)
)
→
(
atleast5
X8
∧
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
=
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
nat_p
(
𝒫
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
¬
exactly5
X7
)
→
exactly4
X7
→
(
(
TransSet
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast6
X7
)
)
)
)
)
∧
(
¬
exactly5
X3
)
)
→
(
¬
atleast5
X6
)
)
→
(
¬
atleast6
X6
)
)
→
(
(
(
TransSet
X3
→
(
¬
exactly2
X3
)
)
∧
(
atleast2
X0
∧
(
(
¬
atleast5
X2
)
∧
exactly3
X3
)
)
)
∧
SNoLt
X4
X1
)
)
)
)
∧
exactly3
X3
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
SNo
X3
)
)
)
)
∧
(
∀X3 :
set
,
(
∃X4 ∈
X2
,
nat_p
X2
)
→
(
∀X4 :
set
,
(
(
¬
exactly5
X3
)
∧
(
(
set_of_pairs
X4
∧
(
¬
atleast4
X1
)
)
∧
(
exactly4
X4
→
atleast4
X3
)
)
)
)
)
)
∧
(
∃X3 ∈
X0
,
∃X4 :
set
,
(
(
atleast5
X0
∧
setsum_p
X3
)
∧
(
(
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
⊆
X4
)
→
atleast4
X4
)
∧
(
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
¬
PNo_downc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
¬
X6
X0
)
)
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
λX5 :
set
⇒
set_of_pairs
X5
)
)
)
)
)
)
)
)
→
(
∀X2 :
set
,
∀X3
∈
X2
,
∀X4 :
set
,
(
(
¬
atleast6
X2
)
∧
exactly3
X4
)
)
)
→
(
∀X2 :
set
,
(
atleast5
X1
∧
(
∀X3 :
set
,
(
(
¬
atleast3
X2
)
∧
exactly4
X1
)
→
exactly5
X2
)
)
→
(
¬
exactly4
X1
)
)
)
)
In Proofgold the corresponding term root is
f0daa4...
and proposition id is
4a768d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMQiDXhTCVnrSXBHVmekCdagS2svJjizXE2
)
∃X0 :
set
,
(
(
X0
⊆
proj0
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
∀X1 :
set
,
∃X2 :
set
,
(
(
X2
⊆
mul_nat
X1
X1
)
∧
(
∀X3 :
set
,
(
∃X4 :
set
,
(
(
equip
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
X3
∧
(
(
atleast2
(
binrep
X3
X4
)
→
atleast2
X4
→
(
¬
atleast3
X4
)
)
→
(
(
(
(
tuple_p
X4
X4
→
(
(
(
exactly2
(
V_
X1
)
∧
atleast5
X2
)
→
(
setsum_p
X3
∧
(
(
(
¬
atleast2
X4
)
∧
(
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
antisymmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
exactly4
(
binunion
X7
X8
)
)
)
)
∧
(
(
¬
nat_p
X4
)
∧
SNoLe
(
ordsucc
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
X4
)
)
∧
(
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast5
X0
)
)
∧
atleast6
X3
)
)
∧
(
(
nat_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
¬
atleast3
X3
)
∧
(
(
(
TransSet
(
ordsucc
X1
)
→
(
(
¬
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
exactly3
X3
)
)
→
ordinal
X4
)
∧
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
atleast5
X4
)
)
)
)
)
∧
ordinal
X4
)
)
)
→
(
(
¬
linear_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
atleast4
X3
∧
(
(
¬
atleast2
∅
)
→
(
(
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
atleast6
X5
)
)
∧
(
¬
exactly4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
∧
(
¬
nat_p
X4
)
)
)
)
)
∧
(
¬
atleast2
X4
)
)
)
→
(
¬
atleast2
X0
)
)
→
TransSet
X4
)
∧
(
¬
exactly4
(
ordsucc
X3
)
)
)
)
)
∧
(
atleast3
∅
→
(
(
(
¬
atleast3
∅
)
→
(
¬
ordinal
X0
)
→
(
¬
SNo
X3
)
)
∧
(
¬
atleast6
X2
)
)
)
)
)
→
(
∀X4
∈
X3
,
atleast5
X4
)
)
)
)
)
In Proofgold the corresponding term root is
178845...
and proposition id is
4f8af8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMLwHSFQaUB3Hx7W8hXJQ6tm4WuKxmknstB
)
∀X0 :
set
,
(
∀X1
∈
V_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
,
∀X2 :
set
,
(
∀X3
∈
∅
,
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
∃X4 :
set
,
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
∃X3 ∈
X1
,
atleast3
X2
)
)
→
(
∀X1
∈
X0
,
(
(
∀X2
⊆
X1
,
(
∀X3 :
set
,
atleast2
X3
→
(
¬
atleast2
X1
)
)
→
(
∃X3 ∈
X0
,
∃X4 :
set
,
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
exactly5
X4
∧
(
exactly4
(
If_i
(
(
(
(
SNoLt
X3
X4
∧
(
¬
atleast3
X2
)
)
∧
(
atleast3
X4
→
(
set_of_pairs
X0
∧
(
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∧
(
¬
atleast2
X3
)
)
)
)
)
→
(
(
(
TransSet
X0
→
(
¬
atleast6
X4
)
)
∧
(
(
X4
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
¬
exactly2
X3
)
)
)
∧
(
(
¬
exactly5
X3
)
→
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
TransSet
X4
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
X2
)
→
(
¬
atleast4
X0
)
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
∃X2 :
set
,
(
(
∀X3 :
set
,
atleast6
X1
)
∧
(
¬
setsum_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
68a0e8...
and proposition id is
4519b4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMNrmQKowNAQnFa6q3vRC2Rr5R1yBULQbX6
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
(
(
¬
nat_p
X1
)
∧
(
∃X2 :
set
,
∃X3 :
set
,
(
(
X3
⊆
X0
)
∧
(
∃X4 :
set
,
(
(
¬
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast4
X6
)
)
∧
(
(
(
¬
atleast4
X0
)
∧
equip
X3
X1
)
→
ordinal
X2
)
)
)
)
)
)
→
(
¬
nat_p
X1
)
→
(
∀X2 :
set
,
(
TransSet
X2
→
exactly5
X1
)
→
(
(
∃X3 :
set
,
∀X4
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
,
exactly1of2
(
¬
exactly3
X3
)
(
nat_p
X3
→
atleast4
X4
)
)
∧
(
∃X3 :
set
,
(
(
∀X4
∈
X3
,
atleast5
X4
)
∧
(
(
(
∀X4
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
,
(
(
¬
SNo
X4
)
∧
(
(
(
¬
exactly2
X3
)
∧
(
¬
atleast2
X0
)
)
∧
(
(
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
exactly5
(
⋃
X3
)
∧
(
(
(
(
(
(
(
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
SNoLt
X6
∅
∧
(
¬
set_of_pairs
X6
)
)
→
(
X3
⊆
X0
)
)
→
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
→
(
¬
atleast5
X3
)
)
→
exactly4
X3
)
∧
(
(
¬
atleast2
X4
)
∧
(
(
¬
atleast5
X2
)
→
(
¬
atleast4
X4
)
→
atleast5
X4
)
)
)
∧
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
¬
atleast3
X3
)
∧
(
¬
atleast2
X4
)
)
)
)
)
)
→
(
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
atleast3
X4
)
)
)
→
(
∀X4 :
set
,
(
¬
atleast2
X0
)
→
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7a76c7...
and proposition id is
9d8121...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVZ7DXgx9MNNfQgScWd3d2jWQT4skiP3gZ
)
∀X0
∈
Sing
∅
,
∃X1 :
set
,
∀X2 :
set
,
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
∀X3
∈
X0
,
∀X4 :
set
,
(
¬
(
X4
⊆
X3
)
)
→
equip
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
X3
→
(
TransSet
X2
∧
(
¬
exactly3
X4
)
)
)
∧
(
∀X3 :
set
,
∃X4 :
set
,
(
(
¬
setsum_p
X3
)
∧
exactly5
X2
)
)
)
In Proofgold the corresponding term root is
147c0b...
and proposition id is
24f8f0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMGRGzitdJRMjtMUfsBzgPf22qu9F5cwJ2A
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
(
∀X2 :
set
,
∃X3 :
set
,
(
(
∀X4 :
set
,
(
¬
atleast5
(
binunion
X3
X4
)
)
→
(
¬
exactly4
X4
)
)
∧
(
∃X4 ∈
X3
,
exactly2
X1
→
(
atleast6
X4
∧
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
¬
exactly3
X4
)
)
)
→
(
SNo_
(
Inj1
X2
)
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
(
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
setsum_p
X4
)
)
)
)
→
(
∃X4 :
set
,
(
(
(
¬
(
X4
∈
X3
)
)
→
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
(
(
¬
nat_p
X3
)
∧
(
¬
exactly3
X4
)
)
∧
(
(
¬
setsum_p
X1
)
∧
(
atleast3
X3
→
(
¬
exactly3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
)
)
∧
(
∀X2
⊆
∅
,
∃X3 :
set
,
(
(
∃X4 :
set
,
(
(
¬
exactly2
X0
)
∧
(
(
¬
SNo
X2
)
∧
(
(
(
(
(
(
exactly2
X3
∧
(
SNoLt
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
(
⋃
X2
)
→
atleast3
∅
→
exactly2
X3
→
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X3
)
)
)
)
∧
TransSet
∅
)
∧
(
(
X2
∈
X3
)
→
exactly3
∅
)
)
→
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
exactly3
X4
)
∧
(
(
(
(
¬
atleast6
X3
)
→
(
¬
atleast4
(
Pi
X2
(
λX5 :
set
⇒
X4
)
)
)
)
→
(
(
setsum_p
X4
∧
(
¬
ordinal
X2
)
)
∧
(
(
(
¬
atleast5
X2
)
→
TransSet
X0
)
→
SNo_
X3
X0
)
)
)
∧
SNo
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
∀X4 :
set
,
(
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
setsum_p
X2
∧
(
(
atleast4
X2
→
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
exactly4
X4
)
)
)
)
)
∧
(
∀X1
∈
X0
,
(
∃X2 :
set
,
(
(
∃X3 :
set
,
∀X4 :
set
,
(
¬
exactly4
X3
)
)
∧
exactly5
X2
)
)
→
(
¬
atleast5
X1
)
)
)
In Proofgold the corresponding term root is
ad0460...
and proposition id is
7b8305...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMdMRveZy7wUJYeTMY4YLEM3pAys2kTnK9y
)
∀X0
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
∃X1 :
set
,
(
(
atleast6
∅
→
(
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
(
∃X3 :
set
,
(
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
∀X4
∈
X2
,
(
¬
SNoLt
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
X0
)
)
)
)
)
)
)
∧
(
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∃X3 :
set
,
(
(
∃X4 ∈
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
atleast6
X3
)
∧
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
→
(
∀X2
⊆
X0
,
∀X3
∈
X2
,
∃X4 ∈
X2
,
(
atleast3
X4
∧
(
¬
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
b41b46...
and proposition id is
ea3866...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYeaSSwNJXtruA4Pc2oMUzyFq9jDp3KFVV
)
∃X0 :
set
,
(
(
∀X1
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
(
(
(
(
¬
atleast6
∅
)
→
(
exactly3
X1
∧
(
(
∀X2
⊆
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
exactly5
X1
)
∧
(
(
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
∀X4
⊆
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
,
(
(
(
(
(
(
¬
atleast4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
nat_p
(
Sep2
X4
(
λX5 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
(
λX5 :
set
⇒
λX6 :
set
⇒
TransSet
(
Sing
∅
)
)
)
)
∧
(
(
exactly4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
SNoLe
∅
X4
)
)
∧
(
¬
ordinal
(
lam2
(
PSNo
X0
(
λX5 :
set
⇒
(
atleast5
X5
→
(
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
(
exactly5
X5
→
(
(
¬
exactly4
X3
)
→
tuple_p
X5
X5
)
→
(
(
atleast3
∅
∧
(
(
¬
atleast2
(
⋃
X5
)
)
∧
(
¬
atleast6
X1
)
)
)
∧
(
(
X5
⊆
X4
)
∧
(
¬
exactly2
X4
)
)
)
→
(
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
(
¬
atleast2
X3
)
→
(
(
(
(
¬
ordinal
X3
)
∧
(
(
(
(
¬
exactly5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
ordinal
X1
)
)
∧
(
(
(
(
atleast6
X2
→
(
(
(
(
(
(
(
(
(
(
¬
atleastp
X4
X5
)
→
nat_p
X2
)
∧
(
¬
setsum_p
X3
)
)
∧
(
eqreln_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
tuple_p
X6
(
⋃
X7
)
)
)
∧
exactly4
(
binintersect
X5
X4
)
)
)
∧
(
¬
exactly2
∅
)
)
∧
(
¬
atleast6
X5
)
)
→
(
(
atleast2
X1
∧
(
¬
exactly2
X1
)
)
∧
(
¬
TransSet
X4
)
)
)
→
TransSet
X4
→
(
¬
atleast2
X3
)
)
→
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
¬
exactly2
X0
)
)
)
∧
exactly4
X4
)
→
(
¬
atleast6
X5
)
)
→
(
(
(
(
setsum_p
X5
∧
(
(
exactly5
X4
→
SNo
X0
)
→
(
¬
atleast2
X0
)
)
)
∧
(
¬
exactly3
X5
)
)
→
atleast3
X1
→
exactly5
X5
)
∧
(
(
(
atleast4
X4
→
(
¬
exactly5
X4
)
→
(
(
¬
nat_p
X5
)
∧
(
(
¬
exactly4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
atleast5
X4
)
)
)
→
(
(
exactly3
X4
→
(
TransSet
X5
∧
(
¬
atleast6
X2
)
)
)
∧
atleast3
X4
)
)
→
(
(
¬
atleast5
X1
)
∧
(
¬
SNo
X2
)
)
)
)
)
)
∧
SNo
X0
)
)
→
(
(
(
tuple_p
X5
X4
∧
(
(
(
¬
(
X0
=
X5
)
)
∧
(
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
¬
atleast5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
ordinal
X4
)
)
)
→
(
¬
atleast4
X4
)
)
)
→
ordinal
X5
→
(
(
(
(
(
¬
nat_p
∅
)
∧
atleast5
X5
)
→
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
nat_p
X1
)
)
∧
(
atleast6
∅
∧
(
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
¬
SNo_
X5
X4
)
→
(
(
atleast2
X4
∧
(
¬
tuple_p
X5
X5
)
)
∧
(
¬
atleast5
X4
)
)
)
)
)
→
(
(
¬
exactly3
X4
)
∧
(
¬
exactly4
X2
)
)
)
)
→
atleast6
X1
→
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast6
X1
)
)
∧
atleast3
X5
)
)
∧
(
(
exactly5
X4
→
atleast5
X5
)
∧
(
(
(
atleast2
X4
∧
(
(
X5
∈
X3
)
∧
setsum_p
X5
)
)
→
exactly2
X5
→
(
¬
TransSet
X5
)
)
∧
(
(
(
exactly3
X5
∧
(
¬
(
SNoElts_
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∈
X4
)
)
)
∧
(
(
symmetric_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
¬
(
X6
∈
X6
)
)
∧
(
¬
atleast5
X7
)
)
→
equip
X7
X7
)
∧
(
(
¬
atleast5
X5
)
→
(
¬
atleast4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
ordinal
X4
)
)
)
→
(
¬
nat_p
X5
)
→
(
(
(
¬
atleast2
X2
)
→
(
exactly2
∅
∧
(
¬
exactly4
∅
)
)
)
∧
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
X4
⊆
X2
)
)
)
)
)
)
)
)
→
(
exactly4
X5
∧
TransSet
X3
)
)
∧
(
(
exactly5
X5
→
(
exactly2
X4
∧
atleast5
X4
)
)
→
(
(
(
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
nat_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
atleast2
X4
→
exactly2
X0
→
nat_p
(
binintersect
X0
X4
)
→
atleast5
X4
→
(
¬
ordinal
X4
)
)
)
∧
(
¬
ordinal
X3
)
)
)
)
)
)
→
exactly5
∅
)
)
(
λX5 :
set
⇒
X4
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X2
)
)
)
)
)
∧
(
¬
atleast6
X1
)
)
→
atleast4
X0
)
∧
(
¬
SNoLt
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X0
)
∧
(
¬
exactly4
∅
)
)
)
)
)
∧
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
¬
atleast3
(
Sing
X2
)
)
)
)
)
)
→
(
∃X2 :
set
,
∀X3
∈
X1
,
(
(
∀X4 :
set
,
(
¬
ordinal
X3
)
→
(
atleast3
X4
∧
(
(
¬
ordinal
X4
)
→
(
(
∅
⊆
X4
)
∧
(
¬
ordinal
X2
)
)
)
)
)
∧
(
∃X4 ∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
exactly2
X2
)
)
)
)
)
)
)
∧
(
¬
exactly5
X0
)
)
∧
(
∃X2 :
set
,
(
(
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
SNoLe
X1
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
(
∃X3 ∈
X0
,
(
X0
∈
X3
)
)
∧
(
(
(
¬
setsum_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
atleast4
X2
)
∧
(
∀X3
∈
X1
,
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
¬
PNoLe
X0
(
λX5 :
set
⇒
(
(
(
¬
ordinal
X3
)
∧
(
(
¬
atleast2
X5
)
∧
(
¬
atleast6
X5
)
)
)
∧
(
¬
atleast6
X2
)
)
)
∅
(
λX5 :
set
⇒
setsum_p
X5
)
)
→
(
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
(
(
¬
atleast5
∅
)
∧
(
¬
atleast3
X4
)
)
→
atleast2
X4
)
→
(
¬
atleast3
X2
)
→
atleast5
X3
)
)
)
)
)
)
)
)
)
)
)
∧
(
∀X1
∈
setprod
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
,
(
∃X2 :
set
,
∀X3
⊆
X1
,
exactly4
X2
)
→
(
(
∀X2
⊆
X1
,
∀X3
⊆
X1
,
∀X4
∈
∅
,
(
exactly5
X0
∧
(
X2
=
X0
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
¬
exactly3
X2
)
)
)
)
)
)
In Proofgold the corresponding term root is
803d0e...
and proposition id is
0732eb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMREXQphuHq9Dvfg2xNWBvgSUGbgkuSnw5W
)
∃X0 :
set
,
(
(
∀X1
⊆
X0
,
∃X2 :
set
,
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
∀X1 :
set
,
∃X2 :
set
,
(
(
X2
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
∀X3 :
set
,
(
(
¬
exactly2
X2
)
∧
(
¬
atleast4
X0
)
)
→
(
¬
exactly5
(
UPair
X2
X2
)
)
)
)
)
)
In Proofgold the corresponding term root is
52e884...
and proposition id is
c738be...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMTjkNWK8NAMcwjKeVU1h6xcJ6dMs6LN8dT
)
∀X0
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
,
∀X1 :
set
,
(
(
∃X2 ∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
,
∀X3 :
set
,
(
∃X4 ∈
X3
,
atleast6
X3
)
→
(
∃X4 :
set
,
(
(
atleast3
(
V_
X2
)
→
(
(
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
set_of_pairs
X4
)
∧
(
¬
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
set_of_pairs
∅
)
∧
(
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
¬
exactly4
X5
)
)
→
(
(
¬
SNoLe
X6
X5
)
∧
(
¬
atleast5
X2
)
)
)
)
)
)
)
)
∧
(
(
(
¬
exactly5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast6
X2
)
→
(
¬
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∈
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
(
(
(
(
¬
atleast2
X2
)
→
(
(
¬
exactly5
X3
)
∧
atleast2
X1
)
)
→
(
¬
exactly3
X4
)
)
∧
(
(
(
¬
atleast3
X2
)
∧
(
¬
exactly2
X4
)
)
∧
(
(
¬
atleast3
X4
)
→
(
¬
atleast3
X4
)
)
)
)
→
(
¬
atleast6
X3
)
)
)
)
)
→
ordinal
X0
)
→
(
∀X2 :
set
,
(
(
∃X3 :
set
,
(
(
X3
⊆
∅
)
∧
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
nat_p
X1
)
→
(
∃X3 ∈
X0
,
∃X4 ∈
Unj
X3
,
inj
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
binunion
X1
X4
)
(
λX5 :
set
⇒
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
In Proofgold the corresponding term root is
b071ef...
and proposition id is
176fbf...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMRUFsZWoiozZr1cMsZUie61yhWWcVs5hML
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
∀X1
∈
UPair
X0
X0
,
∀X2 :
set
,
∀X3 :
set
,
(
(
(
¬
atleast2
∅
)
∧
(
∀X4 :
set
,
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
¬
exactly5
X3
)
→
(
exactly5
(
Inj1
X4
)
∧
(
(
¬
set_of_pairs
X3
)
∧
(
¬
set_of_pairs
X4
)
)
)
→
(
(
exactly4
X4
∧
(
(
¬
atleast4
X4
)
→
exactly5
X4
)
)
∧
(
¬
exactly5
X4
)
)
)
)
∧
(
(
(
(
¬
setsum_p
(
setsum
X0
X0
)
)
∧
(
¬
atleast6
X0
)
)
→
(
∀X4 :
set
,
(
¬
atleast5
(
ordsucc
X2
)
)
→
(
¬
nat_p
X2
)
)
)
→
atleast6
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
In Proofgold the corresponding term root is
27a017...
and proposition id is
f797ef...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMRwYWsqPf79Ubk6dYfLXVwXGu1yvJMDyB5
)
∃X0 :
set
,
(
(
∃X1 :
set
,
(
nat_p
X0
∧
(
∃X2 ∈
X1
,
∃X3 :
set
,
(
(
X3
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
(
(
∀X4 :
set
,
(
¬
atleast3
X0
)
→
(
¬
ordinal
∅
)
→
(
¬
atleast4
X3
)
)
→
(
∀X4
⊆
X1
,
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
∃X4 :
set
,
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
∧
(
∃X1 :
set
,
(
(
X1
⊆
∅
)
∧
(
∀X2 :
set
,
(
∃X3 ∈
X2
,
∃X4 :
set
,
(
TransSet
X3
∧
TransSet
X2
)
)
→
setsum_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
ordinal
∅
→
(
∃X3 :
set
,
(
(
∀X4
∈
X2
,
(
¬
atleast6
X1
)
)
∧
(
(
∃X4 ∈
X2
,
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
exactly3
X4
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ffe783...
and proposition id is
7856fd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMdJRMR59W1AcGaHbAjpQL9jE2u9MQAgVuW
)
∀X0 :
set
,
∀X1
∈
X0
,
(
∀X2
∈
X0
,
∀X3 :
set
,
(
∀X4 :
set
,
(
¬
TransSet
X2
)
)
→
(
∃X4 ∈
∅
,
(
(
¬
linear_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
SNo
X5
)
→
(
(
¬
nat_p
(
Inj0
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
(
¬
linear_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
atleast3
∅
∧
(
(
(
(
SNo
X7
→
set_of_pairs
X2
)
→
(
atleast2
X8
∧
(
atleast5
X2
∧
nat_p
X8
)
)
)
∧
(
exactly3
X5
→
(
(
¬
nat_p
X2
)
∧
(
nat_p
X7
→
atleast6
X8
→
(
¬
nat_p
X6
)
)
)
→
(
(
TransSet
X7
→
exactly4
∅
)
∧
atleast5
X8
)
)
)
∧
(
(
atleast5
X8
→
partialorder_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
¬
exactly3
(
Repl
X10
(
λX11 :
set
⇒
X11
)
)
)
)
→
(
(
atleast5
X0
∧
(
(
X8
∈
X5
)
→
(
(
¬
atleast5
X8
)
∧
(
atleast3
(
proj1
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
exactly4
∅
∧
(
(
¬
TransSet
X2
)
→
(
(
(
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
nat_p
X8
)
→
exactly4
X8
)
∧
nat_p
X7
)
→
(
¬
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
exactly4
X8
∧
(
TransSet
X5
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
X7
⊆
X4
)
)
)
)
)
)
∧
(
¬
atleast3
X0
)
)
)
∧
(
¬
exactly5
X7
)
)
)
)
→
(
(
(
¬
ordinal
X7
)
→
binop_on
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
(
λX9 :
set
⇒
λX10 :
set
⇒
X5
)
)
∧
(
(
(
(
¬
exactly5
X8
)
→
exactly2
X7
)
∧
(
(
¬
(
X7
∈
SNoElts_
X7
)
)
→
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
(
X3
∈
X3
)
→
(
¬
atleast2
X7
)
)
→
(
atleast4
X8
∧
(
(
exactly2
X7
→
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
atleast2
∅
)
)
)
)
)
)
→
atleast5
(
V_
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
X6
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
(
(
(
(
(
atleast3
X5
→
(
¬
atleast4
(
Unj
X6
)
)
)
→
(
¬
atleast2
X2
)
→
(
(
(
¬
setsum_p
X5
)
∧
(
(
(
(
¬
atleast6
X5
)
→
setsum_p
X2
)
∧
exactly2
X6
)
∧
exactly4
X5
)
)
∧
(
(
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
SNo
X5
∧
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
exactly5
X5
)
)
)
)
)
∧
exactly3
X0
)
∧
(
(
SNo
∅
∧
(
(
(
¬
set_of_pairs
X6
)
→
(
(
(
¬
nat_p
X5
)
∧
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
¬
set_of_pairs
X3
)
)
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
set_of_pairs
X0
)
)
∧
(
(
set_of_pairs
X5
∧
(
¬
PNo_upc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
TransSet
X4
)
X5
(
λX7 :
set
⇒
(
(
(
atleast4
X0
→
per_i
(
λX8 :
set
⇒
λX9 :
set
⇒
exactly4
X5
)
)
→
(
¬
atleast2
X5
)
)
∧
exactly3
X6
)
)
)
)
→
(
¬
atleast5
X6
)
)
)
∧
exactly5
X6
)
)
→
(
¬
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
¬
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
(
X3
∈
X3
)
)
)
→
(
∀X2
∈
X1
,
(
∃X3 :
set
,
(
(
∃X4 ∈
X2
,
exactly2
X0
)
∧
(
∀X4 :
set
,
(
(
¬
(
∅
⊆
Sing
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
¬
exactly2
X3
)
)
)
)
)
→
(
(
∃X3 :
set
,
(
(
(
∀X4 :
set
,
(
(
(
(
(
(
atleast5
∅
∧
(
(
(
¬
PNoEq_
X3
(
λX5 :
set
⇒
(
¬
exactly2
∅
)
→
(
¬
ordinal
X3
)
)
(
λX5 :
set
⇒
(
(
(
(
(
¬
atleast3
X5
)
∧
(
¬
nat_p
X2
)
)
∧
(
(
¬
atleast5
X5
)
∧
(
(
(
(
¬
exactly5
X5
)
→
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
(
(
atleast3
X5
∧
(
(
(
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
(
¬
exactly4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
atleast2
(
Inj1
X1
)
)
)
∧
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
atleast5
X4
)
)
→
(
¬
atleast6
X1
)
→
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
exactly5
X1
→
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
atleast3
X4
)
)
∧
ordinal
X5
)
∧
(
¬
nat_p
X5
)
)
∧
SNo
X4
)
)
∧
exactly3
X2
)
)
)
→
(
¬
nat_p
X4
)
)
→
(
(
(
¬
atleast6
X4
)
→
(
¬
tuple_p
X2
∅
)
)
→
(
¬
partialorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
(
(
atleast3
X7
∧
(
atleast3
(
𝒫
X2
)
∧
(
¬
ordinal
X1
)
)
)
→
(
(
(
exactly3
X4
→
(
exactly4
X6
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
∧
(
atleast6
∅
∧
atleast5
∅
)
)
∧
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
ordinal
X8
)
)
)
)
)
∧
exactly3
X6
)
)
)
)
→
(
X1
=
X0
)
)
→
(
¬
atleast4
X2
)
)
)
→
(
¬
exactly4
X3
)
)
→
(
¬
tuple_p
X4
X4
)
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
atleast4
X4
∧
(
¬
atleast6
(
SetAdjoin
X3
X4
)
)
)
)
∧
TransSet
X3
)
∧
(
¬
exactly2
X3
)
)
∧
(
exactly3
X2
→
(
¬
ordinal
(
binunion
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
X4
)
)
)
)
→
(
(
(
¬
atleast2
X3
)
→
(
¬
atleast4
X4
)
)
∧
(
(
(
ordinal
X2
∧
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
⊆
X4
)
)
→
(
¬
(
X3
=
∅
)
)
→
(
¬
nat_p
X0
)
→
(
(
exactly2
X2
→
(
¬
(
X4
∈
∅
)
)
)
∧
atleast6
X2
)
)
∧
(
¬
exactly4
X4
)
)
)
)
→
(
∃X4 :
set
,
exactly2
(
Pi
∅
(
λX5 :
set
⇒
X4
)
)
)
)
∧
(
(
(
∃X4 :
set
,
(
(
(
(
atleast3
X4
→
(
¬
exactly3
X0
)
)
→
(
(
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
tuple_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
X2
∧
(
¬
tuple_p
X4
X4
)
)
)
∧
(
(
(
atleast6
X2
→
(
(
atleast2
X4
→
(
ordinal
X4
∧
(
¬
equip
X4
X0
)
)
)
∧
setsum_p
X3
)
)
∧
atleast2
X3
)
∧
(
X3
∈
X2
)
)
)
→
(
¬
setsum_p
X4
)
)
∧
(
¬
SNo
X1
)
)
∧
(
(
(
¬
exactly5
X2
)
→
atleast3
X4
)
→
(
(
exactly4
X4
∧
(
¬
TransSet
X3
)
)
∧
(
(
(
PNoEq_
X4
(
λX5 :
set
⇒
(
exactly2
X3
→
(
(
(
(
¬
atleast6
X5
)
∧
exactly4
X5
)
∧
(
atleast3
∅
∧
(
X4
∈
SNoElts_
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
→
(
(
(
exactly4
X4
→
(
(
tuple_p
X5
X0
→
exactly2
X5
)
∧
(
¬
exactly5
X4
)
)
)
→
(
¬
exactly5
X4
)
)
∧
(
(
exactly4
X0
→
exactly5
X5
)
→
(
(
(
(
(
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
set_of_pairs
∅
)
)
→
(
¬
set_of_pairs
X4
)
)
∧
(
exactly3
X4
∧
(
¬
atleast2
X2
)
)
)
∧
exactly5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
(
λX5 :
set
⇒
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
atleastp
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
(
(
(
¬
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
atleast3
∅
)
→
(
¬
exactly4
∅
)
)
→
exactly3
X6
)
)
∧
(
atleast2
X2
→
atleast4
∅
)
)
∧
(
¬
exactly3
X4
)
)
→
(
(
¬
atleast4
X3
)
∧
(
¬
setsum_p
X4
)
)
)
∧
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
atleast2
X4
)
)
)
)
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
¬
set_of_pairs
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
∧
TransSet
X3
)
)
)
→
(
∀X3 :
set
,
(
∀X4
∈
X2
,
linear_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
TransSet
X4
)
)
)
→
(
∀X4 :
set
,
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
(
∀X3 :
set
,
∀X4
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
,
(
¬
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly3
∅
)
)
)
)
In Proofgold the corresponding term root is
a1f548...
and proposition id is
521725...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMS8NB7tnwudKx67dxN9YKoBeqtFxRAJPZg
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
∀X1 :
set
,
(
∃X2 :
set
,
nat_p
X2
)
→
(
(
∀X2 :
set
,
(
∀X3 :
set
,
(
¬
atleast5
X1
)
)
→
(
∀X3 :
set
,
∀X4
∈
X2
,
setsum_p
X2
)
)
∧
(
(
¬
symmetric_i
(
λX2 :
set
⇒
λX3 :
set
⇒
∀X4
⊆
binunion
X2
X2
,
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
¬
(
X0
⊆
X1
)
)
∧
(
∃X2 :
set
,
(
atleast6
X0
∧
(
∃X3 :
set
,
∃X4 ∈
X2
,
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
exactly4
X0
)
→
(
¬
nat_p
∅
)
→
(
(
¬
atleast4
X5
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
362ba2...
and proposition id is
473877...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYXUPQjhRehxqq6sx1MSzgoN8azJABGN62
)
∃X0 :
set
,
∀X1 :
set
,
(
∀X2
∈
X0
,
(
∀X3 :
set
,
(
¬
PNo_downc
(
λX4 :
set
⇒
λX5 :
set
→
prop
⇒
(
(
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
SNo
X2
→
(
(
(
(
atleast3
X4
→
(
(
¬
X5
X3
)
∧
(
set_of_pairs
X4
→
(
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
(
X5
X4
→
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
⊆
X4
)
)
∧
(
¬
X5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
¬
X5
X1
)
)
→
X5
X4
)
→
atleast5
(
mul_nat
X0
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
X5
X2
)
→
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
X5
X0
)
)
∧
(
¬
X5
X0
)
)
)
X3
(
λX4 :
set
⇒
(
SNo
X2
∧
TransSet
X1
)
→
(
(
(
¬
exactly3
∅
)
∧
(
(
(
TransSet
X0
→
PNoLt_
X3
(
λX5 :
set
⇒
(
¬
atleast6
X4
)
→
atleast6
X4
)
(
λX5 :
set
⇒
(
¬
ordinal
X1
)
)
→
(
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
exactly5
(
proj1
(
SNoElts_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
∧
(
exactly3
X3
→
(
¬
TransSet
(
SNoLev
X3
)
)
)
)
→
atleast6
X2
→
atleast3
X4
)
)
∧
(
¬
exactly3
X4
)
)
)
)
→
atleast2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
(
∀X3
∈
X2
,
∀X4 :
set
,
atleast6
X3
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
∀X3 :
set
,
(
(
∀X4 :
set
,
(
atleast2
X3
∧
(
set_of_pairs
X2
∧
(
¬
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast3
X0
→
atleast3
X5
→
(
(
atleast6
X1
∧
(
(
(
(
(
∅
=
X6
)
→
TransSet
X6
)
→
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
(
¬
SNoLt
∅
X6
)
∧
(
PNo_upc
(
λX7 :
set
⇒
λX8 :
set
→
prop
⇒
(
¬
X8
X7
)
)
X6
(
λX7 :
set
⇒
(
¬
atleast3
X7
)
)
→
(
(
¬
nat_p
∅
)
∧
(
¬
atleast4
X0
)
)
)
)
)
→
(
¬
(
X2
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
atleast3
X6
)
)
)
)
)
→
atleast4
X3
)
∧
(
∃X4 :
set
,
(
(
(
(
exactly2
X3
∧
(
exactly5
X2
∧
exactly3
X0
)
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast3
X0
)
)
)
→
(
(
¬
setsum_p
X3
)
∧
(
(
X4
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
(
atleast4
X1
→
(
¬
exactly2
X3
)
)
∧
(
¬
atleast5
X4
)
)
→
exactly3
X3
→
(
¬
(
X1
∈
X4
)
)
)
→
(
(
(
(
(
¬
setsum_p
X3
)
∧
(
¬
exactly1of2
(
¬
ordinal
(
Sing
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
(
exactly3
X4
→
exactly5
X3
)
)
)
→
(
¬
exactly2
X4
)
→
ordinal
X4
)
→
(
¬
exactly3
X2
)
)
∧
(
(
(
(
(
(
∅
∈
∅
)
→
(
(
SNo
X4
→
(
nat_p
X3
→
(
¬
TransSet
X1
)
)
→
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
atleast5
X3
)
→
atleast5
X2
→
atleast5
X3
)
)
)
→
setsum_p
X3
→
nat_p
∅
)
∧
(
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
(
¬
SNo
X4
)
→
(
X4
=
X0
)
)
)
→
equip
X1
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
∧
(
(
¬
TransSet
X1
)
∧
(
atleast4
X0
∧
(
¬
exactly4
X4
)
)
)
)
∧
(
¬
exactly4
X4
)
)
)
)
→
(
∃X4 ∈
X1
,
(
¬
set_of_pairs
X3
)
→
(
(
(
atleast3
X0
∧
(
(
¬
exactly3
X4
)
→
(
¬
exactly2
X3
)
)
)
∧
(
(
¬
atleast6
X2
)
∧
(
(
(
(
¬
atleast6
X0
)
∧
(
(
¬
exactly3
(
Sep
X0
(
λX5 :
set
⇒
(
¬
TransSet
X4
)
)
)
)
∧
TransSet
X3
)
)
→
(
(
(
(
(
¬
exactly5
∅
)
→
(
(
atleast3
X3
∧
(
¬
exactly2
X3
)
)
∧
atleast2
∅
)
)
→
(
¬
atleast4
X4
)
)
→
atleast5
X3
)
→
(
¬
set_of_pairs
X4
)
)
→
atleast2
X2
)
→
ordinal
X4
)
)
)
∧
(
(
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
SNo
X0
)
→
(
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
SNoLe
X2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
(
¬
exactly2
(
Sing
X4
)
)
∧
(
(
(
(
(
(
¬
atleast3
X3
)
→
(
¬
atleast5
X4
)
)
∧
tuple_p
X1
X3
)
→
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∈
X3
)
)
→
(
ordinal
X4
∧
(
(
(
¬
atleast4
X3
)
→
(
(
nat_p
X0
∧
(
(
¬
SNoLt
X1
X3
)
→
(
¬
exactly4
X4
)
)
)
∧
(
¬
SNoLt
X4
∅
)
)
)
∧
exactly3
X3
)
)
)
→
(
atleast5
X3
∧
(
ordinal
(
lam2
X3
(
λX5 :
set
⇒
X4
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X6
)
)
∧
(
exactly2
X3
∧
ordinal
X4
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
608817...
and proposition id is
64d737...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMQKjLjQEDBJcnr1Tf9JtQPQ8qoGB9G2cpM
)
∃X0 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
∃X1 :
set
,
(
(
∀X2
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
(
∃X3 :
set
,
atleast3
X3
)
→
(
∃X3 :
set
,
(
(
TransSet
X1
→
(
∀X4 :
set
,
ordinal
X2
→
(
¬
TransSet
X4
)
)
)
∧
(
¬
atleast6
X3
)
)
)
)
∧
(
¬
exactly2
X0
)
)
In Proofgold the corresponding term root is
168104...
and proposition id is
47657c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMZ1E3eHsSekrjcjn6MPTi2NMBzpRd1bMBN
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
∃X1 :
set
,
(
(
∀X2 :
set
,
(
∀X3
⊆
X2
,
(
(
¬
atleast4
X2
)
∧
(
∃X4 :
set
,
(
(
(
(
(
partialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
setsum_p
X5
)
∧
(
¬
atleast2
X4
)
)
)
∧
(
(
¬
atleast4
X1
)
→
(
(
(
(
(
¬
nat_p
X3
)
→
(
¬
exactly5
(
SNoLev
∅
)
)
→
(
(
¬
atleast3
X4
)
∧
atleast2
X3
)
)
∧
(
(
¬
atleast5
X3
)
∧
set_of_pairs
X2
)
)
∧
(
(
¬
equip
X0
X3
)
→
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
ordinal
X2
)
→
(
atleast4
∅
∧
exactly5
X0
)
)
∧
(
¬
atleast5
X0
)
)
→
atleast6
X4
)
)
∧
(
(
(
¬
TransSet
X4
)
→
(
exactly3
X3
∧
exactly3
X2
)
)
∧
atleastp
X2
X4
)
)
)
)
∧
(
exactly4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
(
(
atleast6
X2
∧
atleast6
X2
)
∧
(
(
exactly5
X2
→
ordinal
X0
)
∧
(
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
TransSet
X6
)
→
(
(
(
(
(
X6
∈
X6
)
→
(
¬
atleast4
X5
)
→
atleast6
X6
)
→
(
¬
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
nat_p
X5
)
)
)
)
→
(
(
(
exactly3
∅
∧
(
¬
atleast5
X4
)
)
∧
(
(
exactly4
X6
→
(
(
¬
atleast2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
atleast5
X1
)
→
(
¬
atleast2
X5
)
→
(
(
(
(
(
(
(
(
(
(
(
(
¬
equip
X5
X6
)
→
(
¬
ordinal
∅
)
)
→
(
(
(
(
¬
exactly5
X5
)
∧
(
(
¬
atleast4
∅
)
∧
exactly2
X2
)
)
→
(
¬
nat_p
X5
)
)
∧
nat_p
X5
)
)
∧
(
¬
exactly5
X0
)
)
→
(
¬
SNoLt
X0
X6
)
)
→
(
totalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
exactly3
X0
)
∧
(
¬
atleast2
X6
)
)
)
∧
atleast3
X2
)
∧
(
¬
SNo
X5
)
)
∧
(
set_of_pairs
X2
→
atleast4
X2
)
)
∧
(
(
(
(
(
¬
atleast5
X1
)
→
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
(
atleast6
X5
∧
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast6
X5
)
)
→
setsum_p
X6
)
→
(
¬
ordinal
X6
)
)
)
→
(
(
(
(
(
(
atleast3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
setsum_p
X0
)
∧
(
nat_p
(
SNoElts_
(
Inj0
X6
)
)
∧
(
¬
exactly3
X6
)
)
)
∧
exactly4
(
SetAdjoin
∅
X6
)
)
→
(
¬
atleast6
X0
)
)
∧
(
(
atleast2
X6
∧
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
(
(
(
¬
ordinal
X6
)
∧
(
(
¬
exactly2
X6
)
∧
(
¬
ordinal
X6
)
)
)
→
(
¬
nat_p
X0
)
)
→
exactly2
(
Unj
X4
)
)
∧
(
(
¬
exactly3
X6
)
∧
(
(
(
¬
(
X0
∈
X4
)
)
∧
(
(
nat_p
X5
→
bij
X6
(
Unj
X6
)
(
λX7 :
set
⇒
X5
)
)
∧
(
PNoLt
X0
(
λX7 :
set
⇒
(
¬
setsum_p
X6
)
)
X0
(
λX7 :
set
⇒
(
¬
symmetric_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
atleast2
X9
∧
(
¬
atleast5
(
𝒫
X8
)
)
)
)
)
)
∧
nat_p
X5
)
)
)
→
(
¬
atleast3
X5
)
→
(
(
(
¬
set_of_pairs
(
ordsucc
X5
)
)
→
(
(
(
(
¬
nat_p
X5
)
→
(
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
¬
nat_p
X5
)
)
)
→
(
exactly3
∅
∧
(
(
(
(
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
TransSet
X3
)
→
(
¬
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
equip
X7
X7
)
)
)
→
(
¬
nat_p
X0
)
)
→
(
(
¬
exactly4
X5
)
∧
atleast2
X6
)
)
∧
atleast5
X6
)
)
)
∧
(
¬
atleast5
X5
)
)
)
∧
(
¬
exactly3
X0
)
)
→
set_of_pairs
X5
)
)
)
)
)
∧
(
(
¬
ordinal
X5
)
→
(
¬
atleast3
X0
)
)
)
)
∧
(
(
(
(
(
¬
atleast2
X0
)
→
atleast2
X0
)
∧
(
(
¬
TransSet
X5
)
∧
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
(
(
(
(
nat_p
X1
→
atleast3
X0
)
→
(
¬
set_of_pairs
X4
)
)
∧
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
⊆
X6
)
)
∧
(
(
(
(
(
(
(
(
(
¬
nat_p
X4
)
∧
(
(
(
¬
atleast3
X5
)
∧
(
(
¬
(
X5
∈
X5
)
)
→
TransSet
X5
→
exactly4
X3
)
)
∧
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
(
¬
(
X8
=
SNoLev
∅
)
)
∧
(
(
¬
nat_p
X8
)
∧
(
(
¬
exactly3
(
famunion
X0
(
λX9 :
set
⇒
X8
)
)
)
→
(
X2
=
X2
)
)
)
)
→
(
¬
exactly4
X8
)
→
(
reflexive_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
(
(
¬
setsum_p
X2
)
→
SNo
X5
→
(
(
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
exactly3
X2
)
∧
(
atleast4
X10
→
(
¬
atleast4
X0
)
→
(
(
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
¬
exactly2
X9
)
→
exactly5
X9
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
∧
atleast5
X10
)
)
∧
(
(
¬
nat_p
X7
)
∧
(
ordinal
X8
→
(
(
exactly3
X7
→
(
TransSet
X7
∧
(
exactly2
X8
→
TransSet
X8
)
)
→
(
¬
atleast2
X2
)
→
(
¬
atleast6
X5
)
)
→
exactly2
X7
)
→
(
¬
atleast3
X7
)
)
)
)
)
∧
(
(
¬
atleast6
X7
)
∧
atleast5
X3
)
)
∧
(
¬
atleast5
X6
)
)
)
)
)
∧
(
(
exactly3
X3
∧
(
(
(
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
exactly4
X6
)
∧
(
X5
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
(
(
¬
atleast4
∅
)
→
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
TransSet
X5
∧
(
X3
∈
X6
)
)
→
(
(
¬
atleast6
X6
)
∧
(
¬
atleast4
X5
)
)
)
)
)
)
∧
(
¬
SNo
X1
)
)
→
(
¬
ordinal
(
SetAdjoin
X5
X0
)
)
)
→
(
¬
SNoLe
∅
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast4
(
lam2
X6
(
λX7 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
λX7 :
set
⇒
λX8 :
set
⇒
X0
)
)
)
)
∧
atleast2
(
ap
X6
X1
)
)
∧
nat_p
(
setexp
X0
X6
)
)
)
∧
(
(
(
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
exactly4
X0
)
)
→
linear_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
set_of_pairs
X7
)
)
)
→
(
¬
atleast2
X2
)
)
∧
(
¬
tuple_p
X2
X5
)
)
)
)
∧
exactly4
X6
)
)
→
(
atleast4
X6
∧
SNoLe
X3
X6
)
)
∧
nat_p
X0
)
)
)
→
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast4
X8
)
∧
TransSet
X6
)
→
(
¬
atleast5
X5
)
)
→
(
(
(
setsum_p
X1
∧
(
(
(
¬
set_of_pairs
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
exactly2
X5
→
(
(
(
¬
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
SNo
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
atleast3
X4
)
)
→
(
¬
exactly5
X5
)
)
)
∧
(
(
¬
exactly4
(
SNoElts_
(
⋃
X5
)
)
)
→
TransSet
X6
)
)
)
∧
SNo
X6
)
∧
atleast5
∅
)
)
)
∧
ordinal
(
setexp
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
X6
)
)
)
∧
(
¬
exactly3
X6
)
)
)
→
(
¬
atleast6
X1
)
)
→
equip
X5
X5
)
→
(
(
(
(
(
(
(
(
¬
(
X4
∈
∅
)
)
∧
(
exactly2
X3
→
(
¬
setsum_p
X4
)
)
)
∧
exactly2
(
SetAdjoin
X1
X2
)
)
∧
(
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
(
¬
exactly2
X3
)
∧
(
¬
exactly5
∅
)
)
→
setsum_p
∅
)
)
)
→
(
(
¬
(
X4
∈
X3
)
)
∧
atleast4
X4
)
)
→
TransSet
X4
→
(
¬
atleast5
X0
)
)
∧
(
(
¬
ordinal
X0
)
∧
atleast4
X3
)
)
∧
(
¬
nat_p
X2
)
)
)
→
(
¬
atleast6
X3
)
)
∧
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
exactly4
(
V_
X3
)
)
→
exactly4
(
Inj0
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
exactly4
X2
)
→
atleastp
X1
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
ordinal
X1
)
)
)
)
)
→
(
¬
exactly3
X4
)
)
∧
atleast6
X4
)
→
(
exactly4
X4
∧
(
exactly3
X0
∧
(
¬
SNo
X4
)
)
)
)
)
∧
(
¬
exactly3
X4
)
)
∧
(
SNoLt
X3
X4
∧
(
ordinal
X4
∧
(
nat_p
X4
→
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
(
¬
TransSet
X2
)
)
)
)
)
∧
(
¬
exactly2
X4
)
)
)
)
)
→
(
(
(
(
∀X3
⊆
X0
,
(
¬
TransSet
∅
)
)
∧
(
∀X3 :
set
,
∀X4
⊆
X1
,
(
(
¬
atleast5
X3
)
∧
(
¬
atleast2
X3
)
)
)
)
→
exactly4
X1
)
∧
(
∃X3 :
set
,
(
(
∃X4 ∈
X0
,
(
(
(
¬
set_of_pairs
X3
)
∧
(
¬
(
X2
∈
X3
)
)
)
∧
(
(
(
(
(
(
¬
atleast6
X4
)
→
(
¬
nat_p
X3
)
)
→
atleast5
X2
)
∧
(
¬
(
X3
∈
X3
)
)
)
∧
(
¬
nat_p
X2
)
)
∧
(
nat_p
X4
∧
(
(
¬
exactly4
X2
)
∧
(
(
¬
nat_p
X3
)
→
(
¬
atleast3
X4
)
)
)
)
)
)
)
∧
(
∀X4 :
set
,
ordinal
X0
→
atleast5
X0
→
(
(
¬
PNoLt
X4
(
λX5 :
set
⇒
(
(
(
¬
TransSet
X4
)
∧
(
(
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
(
¬
ordinal
X5
)
→
(
(
¬
exactly5
X4
)
∧
(
(
(
(
(
(
¬
set_of_pairs
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
atleast4
X5
→
(
exactly2
X4
→
atleast4
X4
)
→
TransSet
X4
)
)
→
(
¬
exactly5
X4
)
)
→
(
(
(
X5
⊆
X5
)
→
(
(
(
(
(
(
¬
atleast4
X4
)
→
(
(
(
(
(
¬
irreflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
(
¬
TransSet
X0
)
∧
(
¬
exactly3
∅
)
)
)
→
(
¬
exactly3
X5
)
)
→
nat_p
X2
)
∧
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
ordinal
X4
)
→
exactly2
X4
)
∧
(
ordinal
X0
∧
(
(
atleast4
X4
→
(
(
(
¬
set_of_pairs
X5
)
∧
(
¬
exactly3
X5
)
)
→
(
(
(
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
(
(
(
¬
atleastp
X5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast4
X5
)
)
∧
(
SetAdjoin
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
X5
∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
∈
X0
)
)
∧
(
ordinal
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
→
(
(
(
¬
exactly5
∅
)
→
exactly2
X5
→
(
¬
exactly4
X5
)
→
TransSet
X0
)
∧
(
(
¬
nat_p
X4
)
→
(
¬
exactly4
X4
)
→
(
(
(
atleast2
∅
∧
(
(
(
¬
exactly5
(
Repl
X2
(
λX6 :
set
⇒
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
atleast4
X5
)
→
(
(
X3
∈
X0
)
∧
(
(
(
¬
SNoLe
(
SetAdjoin
X2
X5
)
X0
)
→
(
(
exactly5
X5
∧
TransSet
X4
)
→
(
¬
exactly5
X1
)
)
→
(
¬
set_of_pairs
X4
)
)
→
(
¬
atleast2
X5
)
)
)
)
)
→
atleast6
(
ordsucc
X4
)
)
→
exactly2
∅
→
exactly4
X2
)
→
TransSet
X5
)
)
)
)
)
→
(
X2
∈
∅
)
)
→
(
ordinal
X4
∧
(
¬
atleast5
X5
)
)
)
)
)
→
(
¬
(
X3
=
X2
)
)
)
∧
(
(
¬
exactly5
X4
)
∧
(
(
¬
atleast5
X4
)
∧
(
¬
atleast4
X4
)
)
)
)
→
(
¬
atleastp
X5
(
binunion
X4
X4
)
)
)
∧
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
SNoLe
X4
X4
)
∧
exactly5
X5
)
)
)
)
∧
TransSet
∅
)
)
∧
(
(
(
(
(
¬
TransSet
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
(
X5
=
X4
)
∧
(
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
¬
atleast6
X5
)
→
atleast4
X1
)
)
)
→
(
¬
atleast5
X5
)
→
(
(
(
¬
atleast5
X0
)
∧
(
(
(
¬
exactly5
X2
)
→
(
(
(
(
¬
atleast6
X4
)
→
(
¬
atleast5
X3
)
)
→
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
exactly4
X7
)
)
)
∧
(
(
exactly4
X4
∧
(
¬
nat_p
∅
)
)
∧
atleast6
X0
)
)
)
∧
(
¬
TransSet
X4
)
)
)
∧
(
(
(
¬
atleast5
X5
)
→
(
(
¬
nat_p
X1
)
∧
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
¬
exactly2
X3
)
→
(
¬
atleast3
X0
)
)
)
)
→
(
¬
atleast6
X5
)
)
)
)
∧
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
atleast2
X4
)
→
(
(
(
¬
TransSet
X5
)
∧
tuple_p
X0
X5
)
→
atleast2
X4
→
(
(
exactly3
X3
∧
(
(
¬
SNo
X4
)
→
(
(
¬
SNo
X4
)
→
(
(
exactly5
X5
∧
(
¬
atleast4
X5
)
)
∧
exactly3
X4
)
)
→
(
¬
exactly2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
atleast6
X3
→
(
(
¬
exactly5
X2
)
∧
exactly3
X5
)
)
)
)
→
atleast2
∅
)
∧
(
¬
SNo_
∅
(
ordsucc
(
mul_nat
X2
(
Sing
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
)
X0
(
λX5 :
set
⇒
(
exactly2
∅
∧
(
X0
∈
X4
)
)
)
)
∧
(
nat_p
X4
∧
(
equip
X2
X3
∧
(
¬
atleast5
X3
)
)
)
)
)
)
)
)
)
∧
(
(
exactly5
X1
∧
(
∀X2
∈
X1
,
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
setsum_p
X4
)
)
)
)
)
∧
PNo_downc
(
λX2 :
set
⇒
λX3 :
set
→
prop
⇒
(
(
∀X4 :
set
,
(
¬
X3
X1
)
→
(
(
set_of_pairs
X0
→
(
nat_p
X2
∧
(
(
(
(
(
¬
X3
X1
)
∧
(
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
nat_p
X1
)
)
∧
(
X3
X2
∧
(
X3
X4
∧
X3
X2
)
)
)
)
→
(
¬
atleast4
X4
)
)
∧
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
X3
X2
∧
X3
X2
)
∧
(
¬
X3
X4
)
)
)
)
)
→
nat_p
X4
)
→
equip
X4
X2
)
∧
(
∃X4 :
set
,
(
(
(
¬
atleast5
X4
)
∧
X3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
atleast4
(
famunion
∅
(
λX5 :
set
⇒
X5
)
)
→
ordinal
X0
)
→
X3
X4
)
)
)
)
)
X1
(
λX2 :
set
⇒
∀X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
atleast6
X4
)
→
(
¬
SNoLt
X4
X4
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
85194d...
and proposition id is
c0f219...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMYvuse7tUteMqLXKZMZqjkSy7icQAagprH
)
∃X0 ∈
∅
,
∃X1 ∈
X0
,
∀X2 :
set
,
(
(
¬
trichotomous_or_i
(
λX3 :
set
⇒
λX4 :
set
⇒
atleast2
X3
→
(
atleast6
X2
∧
(
¬
TransSet
X3
)
)
)
)
∧
(
∀X3 :
set
,
(
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast6
X5
)
)
)
)
→
(
∃X4 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
(
(
∃X3 :
set
,
(
(
X3
⊆
X1
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
(
(
(
(
(
¬
exactly5
X0
)
→
atleast4
X4
)
∧
atleast5
X4
)
∧
(
atleast5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
atleast3
X3
)
)
→
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
TransSet
X4
)
)
)
)
)
)
∧
(
∃X3 :
set
,
∃X4 ∈
X2
,
(
(
(
(
¬
exactly2
X2
)
→
(
¬
exactly3
X3
)
)
∧
(
¬
atleast5
X4
)
)
∧
(
¬
nat_p
X3
)
)
)
)
In Proofgold the corresponding term root is
5acaae...
and proposition id is
ce899c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMbeV54bcBrJc5uDRHtA9bqLov7V8gtNEd9
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
X1
⊆
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
¬
nat_p
X2
)
∧
(
(
∃X3 ∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
(
¬
exactly4
X1
)
)
∧
(
∃X3 ∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
∀X4 :
set
,
(
ordinal
X1
∧
(
(
(
¬
atleast2
X2
)
∧
(
¬
exactly5
∅
)
)
∧
atleast5
X3
)
)
)
)
)
)
)
∧
equip
X1
(
Inj1
X1
)
)
)
)
→
(
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
¬
atleast3
X0
)
)
)
In Proofgold the corresponding term root is
c0245f...
and proposition id is
f8ef37...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMT74giqjpC98xFzk1zFtRJmb31Q2UXh7uF
)
∀X0 :
set
,
(
∃X1 :
set
,
∀X2 :
set
,
(
(
∀X3
∈
X2
,
ordinal
∅
)
∧
(
∀X3
⊆
X0
,
∀X4
⊆
X3
,
(
¬
exactly4
X3
)
)
)
→
(
(
¬
exactly2
X0
)
∧
(
¬
TransSet
X1
)
)
)
→
(
∃X1 ∈
∅
,
∃X2 :
set
,
(
irreflexive_i
(
λX3 :
set
⇒
λX4 :
set
⇒
SNo
X0
→
(
(
set_of_pairs
X0
→
exactly4
X3
)
∧
(
¬
(
X1
∈
X2
)
)
)
→
(
¬
exactly3
X2
)
)
∧
TransSet
(
⋃
X1
)
)
)
In Proofgold the corresponding term root is
3512a7...
and proposition id is
78b098...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMHNzWVF5n5wsEdKdas4R2SJKurNsLkSeSe
)
∃X0 ∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
,
∃X1 :
set
,
(
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
∀X3
∈
X1
,
∃X4 :
set
,
(
(
(
(
(
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
atleast3
X1
)
)
∧
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
atleast3
∅
→
atleast3
X4
)
)
∧
(
exactly2
X3
→
(
¬
atleast6
X2
)
→
atleast6
X3
)
)
∧
(
(
(
(
exactly5
X3
∧
(
¬
nat_p
X3
)
)
→
(
¬
exactly2
X3
)
)
→
exactly5
X4
)
→
ordinal
X2
)
)
)
)
)
→
(
∀X2
∈
SNoElts_
X0
,
∃X3 :
set
,
(
(
∀X4
∈
X2
,
atleast5
X3
)
∧
(
∀X4 :
set
,
atleast3
X1
→
(
¬
equip
X3
X4
)
)
)
)
In Proofgold the corresponding term root is
e8cea9...
and proposition id is
f6b702...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMKjKkoa9HDUoLuuoDizx3bjBG2GraUcqwd
)
∀X0
⊆
SetAdjoin
(
binunion
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
(
V_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
(
setprod
(
⋃
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
(
binintersect
(
⋃
(
ordsucc
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
,
∃X1 :
set
,
(
(
∀X2 :
set
,
(
∃X3 :
set
,
(
(
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
exactly5
X3
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
Sing
X1
)
∧
(
nat_p
X3
∧
set_of_pairs
X3
)
)
)
)
→
exactly4
X2
)
→
(
¬
atleast6
X2
)
)
∧
(
∀X2
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
,
∀X3
∈
X2
,
∃X4 :
set
,
(
¬
SNoLe
X3
X4
)
→
(
(
atleast4
X4
→
exactly3
X0
→
(
(
¬
binop_on
X1
(
λX5 :
set
⇒
λX6 :
set
⇒
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
¬
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
atleast2
(
ordsucc
X1
)
→
(
(
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
TransSet
X2
)
∧
atleast6
X0
)
→
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
exactly5
X1
)
)
In Proofgold the corresponding term root is
118f02...
and proposition id is
fe4f86...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMchgB1ep1m9RSeTmZnW7L3j1Grc6CytVTB
)
∃X0 :
set
,
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
(
X4
⊆
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
(
(
(
(
(
(
¬
SNo
X3
)
∧
atleast4
X4
)
→
(
(
(
(
X3
=
X2
)
∧
(
(
exactly3
X3
→
(
(
(
(
atleast5
(
In_rec_i
(
λX5 :
set
⇒
λX6 :
set
→
set
⇒
X0
)
X3
)
→
(
(
∅
∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
exactly5
X4
)
)
→
(
¬
atleast4
∅
)
)
→
atleast4
X0
)
∧
atleast4
X4
)
)
→
(
¬
atleast2
∅
)
)
)
∧
(
(
atleast4
X3
→
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
TransSet
X3
)
)
)
∧
(
(
¬
atleast6
∅
)
∧
(
(
exactly3
X0
→
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
(
¬
atleast2
X4
)
→
(
¬
nat_p
X3
)
→
(
set_of_pairs
X3
→
(
SNo
X4
→
(
¬
atleast4
X3
)
)
→
(
X3
∈
∅
)
)
→
(
¬
nat_p
X0
)
)
→
(
(
¬
SNo
(
Unj
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
∧
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
atleast6
X4
→
(
(
(
(
(
¬
exactly4
X3
)
→
(
exactly5
X4
∧
(
(
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
exactly4
X3
)
)
→
(
(
(
¬
atleast6
X4
)
→
(
(
atleast2
X4
→
(
¬
exactly4
X4
)
)
∧
(
(
(
¬
atleast5
X4
)
→
(
(
¬
exactly2
∅
)
→
ordinal
X3
)
→
tuple_p
X0
X0
)
→
(
¬
exactly5
X0
)
)
)
)
∧
(
(
¬
setsum_p
X3
)
→
atleast6
X3
)
)
)
)
)
∧
(
(
atleast5
X3
∧
(
(
(
(
(
(
X0
∈
X0
)
→
(
(
atleast2
X1
→
(
(
ordinal
X4
∧
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast4
X6
)
)
)
∧
(
(
¬
nat_p
X3
)
∧
(
¬
setsum_p
X1
)
)
)
)
∧
(
(
(
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
setsum_p
X3
→
(
(
¬
TransSet
X4
)
∧
(
¬
atleast6
X4
)
)
→
atleast4
X3
)
)
→
(
X3
∈
X2
)
)
∧
atleast6
∅
)
→
(
¬
atleast3
X4
)
→
atleast5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
exactly3
X4
)
)
)
→
(
¬
(
X4
=
X0
)
)
)
∧
(
¬
TransSet
X4
)
)
∧
(
(
¬
atleast4
X4
)
→
(
X3
∈
X3
)
)
)
→
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
(
(
(
(
(
atleast4
X4
∧
TransSet
X0
)
→
(
(
(
PNoEq_
X3
(
λX5 :
set
⇒
(
(
¬
exactly4
X0
)
∧
(
¬
exactly4
X4
)
)
→
TransSet
X5
)
(
λX5 :
set
⇒
atleast3
∅
)
→
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
¬
exactly3
X1
)
→
atleast5
X3
)
∧
(
(
¬
exactly1of2
(
(
¬
exactly4
X3
)
→
(
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
atleast5
X4
)
)
(
exactly5
X4
)
)
∧
exactly5
(
proj0
X4
)
)
)
→
atleast4
X4
)
∧
atleast2
X4
)
→
(
transitive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast2
X6
)
)
∧
atleast5
X0
)
)
∧
(
¬
setsum_p
∅
)
)
)
)
∧
(
(
¬
atleastp
X3
X4
)
∧
(
¬
equip
X0
X2
)
)
)
)
→
(
¬
TransSet
X3
)
→
(
(
¬
atleast2
X3
)
∧
(
(
¬
atleast2
X4
)
→
(
¬
nat_p
X0
)
→
(
¬
exactly5
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
∧
atleast3
X3
)
→
exactly4
∅
)
)
)
)
→
(
¬
atleast5
X1
)
)
→
atleast4
∅
)
∧
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
X0
⊆
X6
)
)
)
)
)
)
∧
(
∀X4
⊆
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
,
(
¬
atleast5
∅
)
)
)
)
∧
(
∀X3 :
set
,
(
¬
(
X3
∈
Inj1
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
∃X4 :
set
,
(
atleast3
X3
∧
(
(
¬
atleast6
X1
)
→
atleast5
X3
)
)
)
)
)
)
∧
(
(
(
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
∃X4 :
set
,
(
(
¬
exactly4
X4
)
∧
(
(
(
(
¬
exactly3
X3
)
∧
set_of_pairs
X3
)
→
atleast4
X4
)
∧
atleast6
X3
)
)
)
∧
(
∃X4 ∈
X2
,
atleast5
X4
)
)
)
∧
(
irreflexive_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
(
¬
SNo
X4
)
∧
(
¬
nat_p
X4
)
)
)
∧
(
∀X3 :
set
,
∃X4 :
set
,
(
(
¬
atleast2
X3
)
∧
(
¬
exactly4
X2
)
)
)
)
)
)
→
(
∀X2 :
set
,
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
¬
exactly2
X0
)
)
)
∧
(
(
(
∀X2
⊆
X1
,
∃X3 :
set
,
(
¬
reflexive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
¬
TransSet
X5
)
)
)
)
→
(
∃X2 :
set
,
(
(
X2
⊆
∅
)
∧
(
∃X3 :
set
,
(
SNo
X3
∧
(
∃X4 :
set
,
(
¬
exactly2
X4
)
)
)
)
)
)
)
→
(
∀X2
⊆
X0
,
(
¬
symmetric_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
¬
atleast4
X4
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7f28c0...
and proposition id is
710f0c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMTHxaWtz7koD99x82ikBj39S6kJb8UAoUH
)
∀X0
∈
binunion
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∅
,
∀X1 :
set
,
(
(
¬
nat_p
X0
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
(
(
∃X3 ∈
X0
,
set_of_pairs
X1
)
→
(
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∧
exactly4
X1
)
)
∧
(
∀X3 :
set
,
∃X4 :
set
,
atleast5
X3
)
)
)
)
)
→
(
∀X2 :
set
,
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
∀X2 :
set
,
(
(
¬
nat_p
X0
)
→
(
∀X3 :
set
,
(
∀X4 :
set
,
exactly4
X0
)
→
(
¬
(
X2
∈
X1
)
)
)
)
→
(
∃X3 ∈
X0
,
(
(
∃X4 :
set
,
(
(
(
¬
atleast2
X3
)
→
(
(
∅
∈
X4
)
∧
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
equip
X4
X2
)
)
∧
(
∀X4 :
set
,
(
(
¬
atleast4
X4
)
→
(
¬
exactly2
X3
)
)
→
(
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
¬
atleast6
X0
)
)
→
(
¬
exactly4
X3
)
)
)
→
(
∃X4 ∈
X0
,
(
(
(
(
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
atleast4
X0
)
∧
(
(
(
¬
SNo_
X1
∅
)
∧
(
(
¬
atleast2
X4
)
∧
(
¬
exactly5
X0
)
)
)
∧
(
(
(
¬
atleastp
X4
X3
)
∧
(
(
(
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
atleast6
X4
)
)
→
(
(
¬
atleast4
X4
)
→
(
SNo
X4
∧
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
→
atleast5
X2
)
∧
setsum_p
X2
)
)
∧
(
SNo
X4
→
(
atleast4
X1
∧
(
(
(
¬
exactly4
X2
)
→
(
(
¬
(
binunion
X2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∈
X2
)
)
∧
(
¬
TransSet
X3
)
)
→
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
∅
=
X3
)
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast4
X0
)
)
)
)
)
→
(
¬
SNoLt
X3
X4
)
)
)
)
)
∧
exactly4
X0
)
∧
(
exactly5
X0
∧
(
¬
atleast6
X3
)
)
)
)
)
)
In Proofgold the corresponding term root is
a49396...
and proposition id is
52b355...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMNZs8juWa2T7vXDEWp5B8wgURaGHDVshQm
)
∀X0 :
set
,
(
∀X1 :
set
,
(
¬
(
X0
∈
X1
)
)
→
(
(
∃X2 :
set
,
(
(
∃X3 :
set
,
(
(
¬
atleast5
X3
)
∧
(
∃X4 ∈
X2
,
(
¬
(
X4
=
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
∧
TransSet
X1
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∀X3
⊆
X1
,
∃X4 ∈
X1
,
(
¬
exactly4
X3
)
)
)
)
)
)
→
(
∃X1 :
set
,
(
(
∀X2 :
set
,
atleastp
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X0
→
(
∀X3
⊆
X2
,
∀X4
∈
X3
,
(
atleast4
X4
∧
(
(
(
(
PNo_downc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
(
(
setsum_p
(
ordsucc
X4
)
→
atleast5
X3
)
∧
exactly4
X4
)
→
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
X6
∅
)
)
)
→
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
X0
(
λX5 :
set
⇒
equip
X5
X1
)
∧
(
¬
nat_p
(
Unj
X0
)
)
)
∧
(
(
X1
=
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
exactly3
∅
)
)
∧
(
(
(
¬
(
X4
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
ordinal
X2
)
∧
(
exactly5
X4
∧
(
(
exactly5
∅
→
atleast6
X4
)
→
(
¬
ordinal
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
¬
SNoLt
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
X3
)
)
)
)
)
∧
(
(
¬
SNo
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
atleast4
X0
)
)
)
In Proofgold the corresponding term root is
287d2c...
and proposition id is
187cc7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMKMRtwCbwL8iXayoUziW2wttmLntbeQGB
)
∀X0
⊆
⋃
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
,
∀X1 :
set
,
∀X2 :
set
,
∃X3 :
set
,
(
(
∀X4
∈
X0
,
(
¬
atleast6
X4
)
→
(
(
(
(
(
¬
tuple_p
X0
∅
)
→
(
(
exactly4
X3
∧
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
∧
SNo
X3
)
)
→
nat_p
X0
)
∧
(
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
(
(
(
¬
exactly3
X3
)
→
atleast6
X4
)
→
TransSet
X1
)
→
atleast2
X3
→
exactly3
∅
)
)
→
(
(
¬
exactly4
X2
)
→
(
¬
exactly4
X3
)
)
→
(
(
(
(
(
¬
atleast2
X2
)
∧
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
→
(
(
(
TransSet
X4
→
(
X0
∈
X3
)
)
∧
(
(
¬
atleast6
X3
)
→
(
(
(
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
(
(
(
(
(
¬
exactly2
(
binintersect
X0
X1
)
)
∧
(
¬
exactly5
∅
)
)
∧
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
exactly5
X3
)
→
(
(
(
exactly2
X0
∧
(
(
(
(
(
(
(
(
¬
exactly2
X4
)
→
(
¬
(
X4
∈
X2
)
)
)
→
(
¬
(
X3
⊆
X3
)
)
)
∧
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
¬
atleast2
X0
)
)
→
(
¬
setsum_p
X3
)
)
∧
(
atleast5
X4
∧
(
(
(
(
¬
equip
X3
X3
)
→
(
¬
exactly3
∅
)
→
(
exactly3
X2
∧
(
(
¬
set_of_pairs
X3
)
∧
(
¬
exactly5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
→
(
¬
ordinal
X4
)
)
∧
(
(
(
(
(
(
(
¬
(
X3
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
nat_p
X3
)
→
(
(
¬
exactly2
X3
)
∧
(
nat_p
X3
→
(
¬
exactly4
∅
)
→
(
(
¬
SNoEq_
X3
X2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
atleast4
X4
)
→
atleast5
X3
)
)
)
)
)
∧
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
atleast4
X3
)
)
→
(
X2
∈
X3
)
)
∧
nat_p
∅
)
)
)
)
∧
(
¬
exactly4
X4
)
)
)
∧
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
(
atleastp
X4
X3
→
(
¬
TransSet
X3
)
→
(
(
¬
exactly4
(
UPair
X4
∅
)
)
∧
(
(
(
(
SNo
∅
→
(
¬
atleast5
X4
)
)
→
(
¬
exactly5
X4
)
)
∧
nat_p
X3
)
→
SNo
X3
)
)
→
(
¬
exactly3
X2
)
)
∧
(
¬
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
(
(
(
¬
ordinal
X0
)
→
ordinal
X4
)
∧
(
(
(
(
(
(
¬
exactly5
X0
)
→
atleast5
X4
→
(
(
(
(
exactly2
X2
∧
(
(
set_of_pairs
X3
→
(
(
¬
atleast4
X3
)
∧
(
¬
exactly3
X3
)
)
)
→
TransSet
(
binunion
X2
X0
)
)
)
∧
(
¬
exactly3
X3
)
)
→
(
atleast6
X3
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly3
X5
)
)
)
→
SNoLe
X3
X3
)
∧
atleast6
(
Inj1
(
𝒫
X4
)
)
)
)
∧
(
(
(
(
TransSet
X4
∧
exactly5
∅
)
∧
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
set_of_pairs
X6
)
)
→
exactly4
X0
)
∧
exactly3
X2
)
)
∧
(
atleast6
X0
→
(
¬
atleast6
X4
)
)
)
∧
(
(
exactly4
X2
∧
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
∧
exactly5
X2
)
)
∧
atleast5
X4
)
)
∧
(
¬
atleast6
X4
)
)
)
∧
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
¬
exactly4
X1
)
)
∧
(
binop_on
X4
(
λX5 :
set
⇒
λX6 :
set
⇒
X0
)
∧
exactly5
X2
)
)
)
)
∧
(
(
(
nat_p
X3
∧
(
exactly2
∅
→
(
(
(
(
(
¬
exactly3
X4
)
→
(
(
¬
nat_p
X3
)
∧
(
¬
ordinal
X3
)
)
)
→
SNo
X1
)
∧
SNo
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
¬
atleast5
X0
)
→
(
(
¬
atleast3
X4
)
∧
(
atleast4
X3
→
(
(
(
(
(
¬
ordinal
X4
)
∧
(
exactly5
X4
∧
(
(
¬
equip
(
Unj
X1
)
∅
)
∧
(
(
¬
(
X3
∈
∅
)
)
→
equip
X4
X4
→
(
(
(
ordinal
X4
→
exactly5
X0
)
→
(
set_of_pairs
X1
∧
(
(
(
(
(
¬
TransSet
X3
)
→
(
(
¬
exactly5
X4
)
∧
(
(
(
atleast6
(
Inj1
X3
)
∧
ordinal
X4
)
→
(
¬
atleast6
X4
)
)
→
(
(
¬
ordinal
X4
)
→
(
(
(
¬
atleast6
X2
)
∧
(
¬
atleast3
∅
)
)
∧
(
(
exactly5
X4
→
(
¬
exactly3
X2
)
)
→
(
atleast6
X3
→
(
(
exactly2
X3
→
atleast2
∅
→
(
¬
SNo
X4
)
)
∧
(
¬
TransSet
X4
)
)
)
→
(
(
(
¬
equip
X4
X3
)
∧
SNo_
∅
X3
)
∧
atleast4
X2
)
)
)
)
→
atleast2
X4
→
(
(
(
exactly3
X0
→
exactly5
X3
)
∧
(
(
¬
nat_p
X4
)
→
atleast4
X4
)
)
∧
(
¬
atleast6
X2
)
)
)
)
)
∧
(
X3
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
(
(
ordinal
X4
→
(
(
¬
atleast5
∅
)
→
(
¬
atleast6
X3
)
)
→
(
¬
ordinal
X4
)
)
∧
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
exactly3
X2
)
)
∧
antisymmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
setsum_p
X1
)
)
)
)
)
∧
(
(
¬
set_of_pairs
X2
)
∧
(
(
¬
ordinal
X2
)
∧
(
exactly5
X1
→
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
nat_p
X2
∧
(
¬
exactly3
X6
)
)
→
(
(
(
¬
exactly4
X0
)
∧
(
(
atleast6
X5
∧
(
¬
TransSet
X6
)
)
→
(
∅
=
X6
)
)
)
∧
(
(
(
(
(
(
¬
per_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast5
X1
)
)
→
(
¬
atleast2
X5
)
)
→
(
¬
exactly4
∅
)
)
∧
(
(
TransSet
(
SNoElts_
X5
)
∧
(
X1
∈
X6
)
)
∧
(
(
(
(
(
¬
partialorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
ordinal
X0
∧
(
¬
SNo
∅
)
)
→
atleast6
X7
→
(
(
¬
set_of_pairs
X2
)
∧
(
SNo
X2
→
(
(
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
atleast6
X0
)
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
(
(
¬
atleast5
X8
)
→
(
¬
(
X7
∈
X7
)
)
)
→
exactly5
(
⋃
X4
)
)
→
(
(
¬
atleast2
X7
)
∧
(
¬
TransSet
X6
)
)
)
∧
(
(
(
¬
exactly1of3
(
atleast3
X8
)
(
exactly2
(
proj1
X8
)
)
(
X2
∈
X7
)
)
→
(
(
(
(
¬
SNo_
X4
X8
)
→
per_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
atleast2
X6
∧
(
¬
TransSet
X10
)
)
)
)
∧
(
¬
SNoLt
X5
X7
)
)
∧
(
¬
exactly3
X4
)
)
)
→
(
exactly5
X8
∧
(
(
(
X7
∈
X7
)
∧
(
set_of_pairs
X8
→
(
(
(
¬
atleast2
(
proj0
X4
)
)
∧
atleast4
X7
)
∧
(
(
¬
atleast6
X8
)
→
atleast3
X7
)
)
)
)
→
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
set_of_pairs
(
Sep
X8
(
λX9 :
set
⇒
(
(
¬
equip
∅
X9
)
∧
(
¬
atleast5
X6
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
ordinal
X8
)
→
(
(
atleast2
X2
→
(
antisymmetric_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
¬
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
¬
exactly3
∅
)
)
→
(
¬
atleast6
X6
)
)
→
atleast2
X8
→
(
(
nat_p
X8
∧
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
exactly2
X7
)
)
∧
atleast5
X7
)
)
→
(
¬
atleast4
X8
)
)
→
atleast6
X1
)
→
(
(
(
¬
(
X8
∈
X1
)
)
→
(
¬
atleast3
(
ap
X7
X4
)
)
)
∧
(
nat_p
X1
→
(
¬
atleast3
(
V_
X1
)
)
→
(
¬
exactly2
X4
)
)
)
)
)
→
atleast3
∅
)
→
(
(
(
¬
atleast5
X0
)
∧
(
(
(
¬
(
X5
⊆
X2
)
)
∧
(
¬
setsum_p
X0
)
)
∧
atleast3
X6
)
)
∧
ordinal
X6
)
→
(
(
SNoElts_
(
nat_primrec
X6
(
λX7 :
set
⇒
λX8 :
set
⇒
X8
)
∅
)
=
X1
)
∧
exactly3
X6
)
→
exactly2
X5
)
→
(
¬
atleast3
(
Inj0
X6
)
)
)
→
(
ordinal
X4
∧
(
¬
atleast4
X5
)
)
)
)
)
∧
ordinal
X6
)
∧
(
(
¬
atleast4
X6
)
→
(
¬
exactly3
X6
)
)
)
)
)
∧
(
exactly5
X2
→
(
set_of_pairs
X4
→
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
∈
X3
)
→
(
(
(
bij
X3
X4
(
λX5 :
set
⇒
X0
)
∧
(
(
¬
atleast3
X3
)
∧
(
(
¬
SNo
∅
)
∧
SNo
X2
)
)
)
∧
atleast5
X2
)
∧
(
atleast6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
(
¬
atleast2
X3
)
∧
(
¬
exactly2
X3
)
)
∧
(
X1
∈
X2
)
)
)
)
)
→
exactly5
X0
)
)
∧
(
(
(
¬
atleast5
X4
)
∧
(
(
¬
atleast6
X2
)
∧
nat_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
setsum_p
X3
)
)
)
)
)
)
→
(
(
(
¬
atleast3
X2
)
∧
(
¬
TransSet
X4
)
)
∧
(
(
(
¬
exactly3
(
V_
(
⋃
(
ordsucc
X4
)
)
)
)
→
(
(
(
(
(
(
(
TransSet
X0
∧
TransSet
(
Sing
X2
)
)
→
(
¬
atleast4
X2
)
→
(
(
(
(
¬
atleast3
X1
)
→
atleast3
∅
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
antisymmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
TransSet
X7
)
→
nat_p
X8
)
→
(
(
¬
atleast6
X7
)
∧
atleast6
X4
)
→
(
ordinal
X8
∧
nat_p
X2
)
)
)
)
)
)
∧
atleast4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
atleast3
X2
→
(
¬
exactly3
X3
)
)
→
exactly5
X3
)
→
(
¬
nat_p
X3
)
)
∧
setsum_p
(
Sing
X4
)
)
→
nat_p
∅
)
∧
ordinal
X4
)
)
∧
(
(
SNoLt
X4
X4
∧
exactly5
∅
)
∧
(
¬
exactly4
∅
)
)
)
)
→
(
(
(
¬
atleast6
X4
)
∧
(
exactly3
X3
→
(
atleast5
X4
∧
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
(
¬
nat_p
X5
)
→
(
(
¬
ordinal
X6
)
∧
(
(
¬
exactly2
X6
)
∧
setsum_p
X5
)
)
→
exactly3
X5
)
∧
(
(
exactly4
X6
→
exactly5
X6
)
→
(
X3
⊆
∅
)
)
)
→
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
(
¬
exactly4
X5
)
)
∧
(
¬
atleast6
X0
)
)
)
)
)
∧
(
(
(
(
(
(
SNo
X3
∧
(
(
(
(
¬
atleast6
(
V_
X4
)
)
→
exactly3
∅
)
→
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
nat_p
X0
)
∧
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
set_of_pairs
X8
)
)
)
)
)
∧
(
(
atleast3
(
Repl
X3
(
λX5 :
set
⇒
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
exactly5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast5
X3
→
(
¬
exactly3
X4
)
→
(
(
(
exactly3
X4
∧
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
atleast5
X1
)
)
∧
(
(
¬
(
X3
=
X3
)
)
∧
(
¬
atleast6
X3
)
)
)
)
)
)
∧
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
tuple_p
X3
X2
)
∧
(
¬
tuple_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
X0
)
)
→
(
¬
exactly2
X2
)
)
∧
(
(
(
¬
exactly3
X2
)
→
(
(
(
(
X4
∈
X4
)
∧
(
(
(
(
¬
exactly2
X0
)
∧
(
¬
nat_p
X2
)
)
∧
(
(
(
(
(
¬
exactly5
X0
)
→
(
set_of_pairs
X3
∧
(
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
(
¬
atleast4
X4
)
∧
ordinal
X0
)
→
(
¬
atleast6
X4
)
)
)
)
→
exactly1of3
(
nat_p
X4
→
(
(
(
(
X4
∈
X3
)
∧
exactly5
X3
)
∧
setsum_p
X4
)
∧
(
(
(
(
(
¬
TransSet
X4
)
→
(
(
¬
SNo_
X4
X0
)
∧
(
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
atleast2
X4
→
(
(
(
atleast3
X4
∧
(
¬
atleast4
X4
)
)
∧
(
¬
atleast5
X2
)
)
∧
(
(
(
¬
atleast5
X4
)
∧
(
atleast3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
equip
X6
X5
)
∧
(
¬
ordinal
X2
)
)
∧
atleast5
X0
)
)
)
∧
atleast3
∅
)
)
)
)
)
→
(
¬
ordinal
X0
)
)
∧
(
(
exactly2
X3
→
(
(
(
¬
(
X3
∈
X3
)
)
→
(
(
¬
transitive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
nat_p
X4
)
)
∧
exactly2
X3
)
)
∧
(
(
¬
(
X4
⊆
X3
)
)
∧
(
(
(
(
(
(
(
(
¬
exactly5
X4
)
→
(
¬
atleast2
X3
)
→
(
¬
TransSet
X2
)
)
∧
(
¬
exactly3
X3
)
)
∧
(
(
(
(
(
(
SNoEq_
X3
X2
∅
→
(
(
¬
exactly2
∅
)
∧
(
set_of_pairs
X0
∧
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
set_of_pairs
X3
)
)
)
∧
(
(
(
(
(
¬
(
X2
∈
∅
)
)
→
(
(
¬
atleast5
X2
)
∧
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
→
(
(
(
¬
exactly3
X4
)
∧
exactly4
X3
)
∧
(
SNo
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
setsum_p
X4
)
)
)
→
(
¬
TransSet
(
Inj1
X3
)
)
)
∧
(
atleast5
X2
∧
(
SNoLe
X1
X3
∧
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
)
→
(
¬
binop_on
X2
(
λX5 :
set
⇒
λX6 :
set
⇒
X5
)
)
)
→
(
(
¬
exactly2
X0
)
∧
(
atleast6
X1
∧
(
(
(
(
(
¬
exactly5
X3
)
∧
(
¬
atleast2
X3
)
)
∧
atleast4
∅
)
→
(
atleast2
X4
∧
(
(
¬
exactly4
X4
)
∧
atleast3
X0
)
)
)
→
set_of_pairs
X1
→
atleast4
X2
)
)
)
)
→
TransSet
X4
→
(
(
(
(
(
(
¬
exactly5
X1
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
(
¬
nat_p
X3
)
∧
(
¬
set_of_pairs
X2
)
)
)
∧
(
(
¬
exactly5
X2
)
∧
(
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
(
¬
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
∧
exactly5
X0
)
∧
(
(
¬
atleast2
∅
)
∧
atleast3
(
lam2
X4
(
λX5 :
set
⇒
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
λX5 :
set
⇒
λX6 :
set
⇒
X0
)
)
)
)
)
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
SNoLt
∅
(
setminus
X6
X5
)
)
→
(
¬
ordinal
X3
)
)
∧
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
(
(
X0
⊆
X6
)
→
(
(
(
¬
ordinal
X6
)
→
(
(
(
(
¬
exactly5
∅
)
∧
TransSet
(
V_
X5
)
)
∧
(
(
¬
atleast6
X2
)
→
ordinal
X5
)
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
(
¬
exactly4
X3
)
)
)
∧
(
(
X4
∈
X0
)
∧
(
(
(
¬
setsum_p
X0
)
→
partialorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
PNo_downc
(
λX9 :
set
⇒
λX10 :
set
→
prop
⇒
(
(
¬
atleast3
∅
)
∧
exactly3
X8
)
)
X2
(
λX9 :
set
⇒
exactly5
X9
)
∧
(
exactly3
X8
∧
(
(
(
¬
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
¬
atleast3
X7
)
→
atleast2
X8
→
atleast6
X8
)
)
→
atleast2
X5
→
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
(
¬
atleast3
X5
)
)
∧
(
(
(
(
(
¬
exactly2
X7
)
→
(
¬
set_of_pairs
X3
)
)
∧
(
¬
atleast3
X8
)
)
→
(
exactly2
X8
∧
(
(
(
(
exactly2
X8
→
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
¬
setsum_p
∅
)
)
∧
(
(
(
¬
exactly4
X2
)
→
(
¬
exactly4
X8
)
)
→
exactly5
X7
)
)
∧
(
(
(
¬
exactly2
X8
)
∧
atleast5
X7
)
→
(
¬
set_of_pairs
X8
)
)
)
)
)
→
(
X7
∈
X4
)
)
)
∧
(
exactly2
X7
∧
(
¬
atleast4
X8
)
)
)
)
→
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
¬
exactly3
X5
)
→
(
atleast2
X8
∧
(
(
bij
X7
X7
(
λX9 :
set
⇒
X9
)
∧
TransSet
X7
)
→
(
¬
atleast3
X8
)
)
)
)
→
(
(
(
(
(
¬
atleast6
X1
)
∧
(
(
equip
X8
X7
→
(
(
(
¬
atleast4
X0
)
∧
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast5
X7
)
→
(
(
(
¬
TransSet
∅
)
→
SNo
X7
)
∧
(
(
¬
atleast4
X8
)
→
(
¬
nat_p
X7
)
)
)
)
)
→
(
atleast6
X3
∧
(
nat_p
X5
∧
(
¬
set_of_pairs
X4
)
)
)
)
→
(
¬
atleast6
X7
)
)
→
(
¬
set_of_pairs
X7
)
)
)
∧
atleast5
X8
)
∧
(
(
atleast5
X5
∧
(
(
¬
nat_p
X7
)
→
(
¬
SNo_
X7
X8
)
)
)
→
(
¬
reflexive_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
(
¬
exactly3
X1
)
∧
(
exactly4
X9
∧
tuple_p
X10
X8
)
)
)
)
)
)
∧
(
(
(
¬
atleast4
∅
)
∧
(
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
ordinal
X1
)
)
→
SNoLe
X7
X8
)
)
)
→
(
TransSet
X5
∧
(
(
X7
∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
→
exactly5
X5
)
)
)
∧
(
(
¬
set_of_pairs
X6
)
→
(
(
atleast5
X5
∧
transitive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
exactly2
X7
)
)
∧
(
¬
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
=
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
)
→
atleast3
X6
→
set_of_pairs
X5
)
)
)
)
)
→
(
¬
set_of_pairs
X4
)
)
→
(
(
¬
exactly2
X3
)
→
PNo_upc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
(
¬
atleast4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
(
¬
X6
(
Inj0
X4
)
)
∧
(
(
(
(
exactly2
X4
∧
atleast2
(
Inj1
X4
)
)
→
(
¬
atleast4
X4
)
)
→
(
¬
X6
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
X6
X5
)
)
→
X6
X5
)
)
)
X3
(
λX5 :
set
⇒
exactly5
X4
)
)
→
(
¬
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
(
¬
atleast6
X1
)
∧
per_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
set_of_pairs
X0
)
)
)
→
exactly2
∅
)
)
)
)
)
→
setsum_p
X0
→
SNo
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
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)
∅
)
)
)
)
(
(
atleast4
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binintersect
(
Inj0
(
binrep
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𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
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(
binunion
X0
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
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𝒫
∅
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)
∅
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𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
¬
(
X2
∈
X1
)
)
→
(
(
¬
atleast5
X3
)
∧
(
¬
TransSet
∅
)
)
→
(
¬
exactly5
X3
)
)
)
∧
(
(
exactly4
X2
∧
exactly2
X0
)
→
set_of_pairs
X4
)
)
(
(
(
(
¬
atleast2
X3
)
→
(
(
¬
SNo
X2
)
∧
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
atleast4
X5
)
→
atleast3
X5
)
∧
(
¬
exactly5
X6
)
)
)
)
∧
(
¬
ordinal
X3
)
)
)
)
→
exactly3
X2
)
→
(
¬
ordinal
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
∧
ordinal
X4
)
→
TransSet
X2
)
→
(
¬
SNo
X4
)
)
)
∧
(
X2
∈
X3
)
)
)
→
atleast5
X3
→
(
(
¬
ordinal
X3
)
∧
(
atleast4
X2
∧
(
¬
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
(
(
¬
exactly4
X0
)
∧
(
(
setsum_p
X4
∧
(
(
¬
exactly5
X3
)
∧
(
(
(
atleastp
∅
X3
∧
(
¬
atleast2
X0
)
)
→
(
exactly4
X3
→
(
¬
exactly4
X4
)
)
→
TransSet
X3
→
(
exactly5
X3
→
(
¬
exactly4
(
proj1
X3
)
)
)
→
SNoLe
X3
∅
)
∧
(
(
¬
exactly5
X1
)
→
(
(
(
(
(
(
(
¬
PNo_downc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
(
X6
∅
∧
X6
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
(
(
(
¬
exactly2
X1
)
→
(
¬
atleast4
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
(
X6
X7
∧
X6
X1
)
→
X6
X0
)
∧
(
nat_p
X4
→
(
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
X6
X8
∧
(
(
(
X6
X7
∧
(
¬
exactly2
X0
)
)
→
(
SNoLt
X4
X8
∧
(
¬
exactly2
X0
)
)
)
→
(
¬
TransSet
X8
)
)
)
∧
(
(
(
(
¬
X6
∅
)
∧
atleast6
X7
)
∧
X6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
X6
∅
∧
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
)
→
(
(
¬
X6
X8
)
→
X6
X0
)
→
(
¬
X6
X7
)
)
)
→
(
¬
(
X0
∈
X8
)
)
)
∧
X6
X5
)
)
)
X3
(
λX5 :
set
⇒
(
(
(
¬
PNoEq_
X5
(
λX6 :
set
⇒
(
¬
exactly2
X5
)
)
(
λX6 :
set
⇒
(
(
(
(
exactly4
X3
→
(
¬
atleast6
X5
)
→
(
¬
atleast4
X5
)
)
→
(
¬
exactly4
X5
)
)
→
(
¬
exactly4
X5
)
)
∧
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
→
(
(
atleast3
X0
→
atleast3
X5
)
∧
(
(
¬
nat_p
X5
)
∧
(
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
⊆
X2
)
→
exactly5
X5
)
)
)
)
∧
(
¬
atleast2
X4
)
)
→
(
¬
exactly3
X4
)
)
)
∧
(
(
TransSet
X4
→
(
exactly5
(
setsum
X4
X4
)
→
(
TransSet
(
Inj0
X3
)
→
(
¬
TransSet
X3
)
)
→
(
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
atleast5
X4
→
SNo_
X3
X4
)
)
)
→
exactly5
X2
)
→
(
¬
SNo_
X4
X3
)
)
)
→
(
(
(
(
(
SNo
X4
→
(
(
(
¬
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
exactly5
X4
)
∧
(
(
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
∧
(
atleast3
X2
∧
exactly3
X3
)
)
∧
(
¬
exactly4
X3
)
)
)
)
→
atleast6
X4
)
→
(
¬
PNoLe
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
λX5 :
set
⇒
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
X3
(
λX5 :
set
⇒
(
¬
atleast4
(
lam
X1
(
λX6 :
set
⇒
X6
)
)
)
)
)
)
→
(
(
¬
exactly5
X2
)
→
(
X3
∈
X3
)
)
→
(
(
(
(
¬
nat_p
∅
)
→
(
(
(
(
(
¬
(
X4
=
X3
)
)
→
(
atleast6
X3
∧
(
(
(
atleast3
X4
→
(
¬
atleast2
X3
)
)
→
nat_p
X2
)
→
(
¬
exactly4
X1
)
)
)
)
∧
(
¬
exactly2
X4
)
)
→
(
¬
atleast6
X4
)
)
∧
(
¬
exactly3
X3
)
)
)
→
(
¬
exactly4
X4
)
)
∧
(
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
nat_p
X2
)
)
)
)
)
∧
exactly4
(
⋃
X4
)
)
)
→
(
¬
ordinal
X3
)
)
→
(
¬
(
X4
∈
X1
)
)
)
∧
exactly1of2
(
(
atleast2
X3
∧
(
atleast5
X0
→
TransSet
X4
→
(
¬
TransSet
X3
)
)
)
∧
(
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
atleast6
X1
∧
atleast4
X3
)
)
∧
(
(
(
¬
atleast3
X3
)
→
(
(
(
¬
atleast2
X2
)
→
atleast5
X0
)
∧
TransSet
(
setminus
(
setprod
X4
X4
)
X3
)
)
→
exactly4
X4
)
∧
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
set_of_pairs
X4
∧
(
(
¬
TransSet
X3
)
∧
(
(
(
¬
exactly5
X3
)
→
exactly5
X2
)
→
(
¬
TransSet
X4
)
)
)
)
)
)
(
X1
∈
ap
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
(
binunion
X3
X3
)
)
)
∧
(
¬
nat_p
∅
)
)
)
)
)
)
∧
atleast4
X4
)
)
→
(
¬
atleast2
X4
)
)
)
)
→
(
(
¬
TransSet
X3
)
∧
(
(
(
set_of_pairs
X2
∧
(
(
(
(
¬
exactly5
(
proj1
X4
)
)
∧
(
¬
atleast5
X2
)
)
∧
(
(
∅
∈
X3
)
∧
(
¬
tuple_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
atleastp
X4
X2
)
)
∧
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
tuple_p
X2
X3
)
)
)
)
)
)
)
)
)
∧
(
(
(
(
¬
nat_p
X0
)
→
(
¬
SNoLe
∅
X4
)
→
nat_p
X1
)
∧
exactly4
X3
)
→
(
¬
PNoLt
X1
(
λX5 :
set
⇒
(
¬
exactly4
X0
)
)
X4
(
λX5 :
set
⇒
(
(
(
¬
exactly2
∅
)
→
(
(
(
atleastp
(
⋃
X4
)
X5
∧
atleast4
X1
)
→
(
(
TransSet
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∧
atleast6
X0
)
∧
(
(
(
¬
TransSet
X1
)
→
(
(
X0
=
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
atleast6
X5
)
)
→
(
(
atleast4
X1
→
(
(
¬
atleast3
X1
)
→
(
¬
atleast5
X4
)
)
→
(
¬
atleast5
X2
)
→
(
exactly4
X4
∧
(
(
¬
atleast2
X4
)
∧
(
¬
exactly3
X4
)
)
)
)
∧
(
exactly4
X5
→
(
(
¬
(
∅
∈
X4
)
)
∧
(
¬
atleast5
X2
)
)
)
)
)
∧
(
(
(
(
(
atleast6
X3
→
SNo
X5
)
→
(
¬
SNo
X2
)
)
→
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
(
¬
TransSet
X5
)
∧
(
¬
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
→
(
¬
exactly5
X0
)
)
)
→
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
∧
(
(
¬
atleast6
X2
)
→
(
TransSet
X0
→
(
atleast6
X0
∧
exactly5
X0
)
)
→
atleast3
X5
)
)
)
)
)
)
∧
(
¬
atleast6
X1
)
)
∧
(
(
atleast5
X3
∧
(
(
¬
nat_p
∅
)
∧
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
exactly2
X6
)
)
)
)
∧
(
(
(
¬
atleast3
X2
)
→
(
(
(
(
(
exactly2
(
Inj1
X3
)
→
(
¬
atleast2
∅
)
→
SNoLe
X4
X1
)
→
(
¬
atleast5
X2
)
→
ordinal
X3
)
∧
(
(
ordinal
X1
→
(
(
¬
atleast6
X3
)
∧
(
¬
set_of_pairs
(
Inj1
(
proj0
∅
)
)
)
)
)
∧
atleast2
X2
)
)
→
(
¬
exactly2
X4
)
)
→
(
¬
nat_p
X4
)
→
(
¬
SNoLe
X4
X3
)
)
→
(
(
¬
exactly2
(
SNoElts_
X3
)
)
∧
(
¬
TransSet
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
exactly2
∅
)
)
)
)
)
→
(
(
¬
atleast2
X2
)
∧
SNo
X1
)
)
)
)
)
→
(
(
exactly2
(
Inj1
X3
)
→
exactly3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
atleast6
X4
)
)
→
(
¬
atleast5
∅
)
→
atleast6
X3
)
)
)
→
(
¬
equip
X4
X2
)
)
→
exactly2
X3
)
→
exactly2
X2
)
→
(
X4
∈
X1
)
)
∧
(
∀X4
∈
X3
,
atleast3
X1
→
(
(
¬
exactly5
X0
)
→
(
(
¬
ordinal
X4
)
∧
ordinal
X4
)
→
(
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
(
¬
(
X3
∈
X4
)
)
∧
(
atleast5
X2
→
(
(
(
(
¬
exactly5
(
setprod
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
Sing
∅
)
)
)
∧
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
exactly5
∅
∧
(
¬
exactly2
X6
)
)
)
)
∧
(
¬
exactly3
∅
)
)
)
)
)
→
setsum_p
X4
)
→
atleast2
(
PSNo
X4
(
λX5 :
set
⇒
(
(
¬
(
X5
∈
X4
)
)
∧
(
ordinal
X0
→
exactly4
X4
→
(
(
¬
(
X5
⊆
X4
)
)
∧
(
(
¬
atleast6
X5
)
∧
(
(
(
setsum_p
X1
∧
(
¬
(
X5
=
X2
)
)
)
→
(
(
¬
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
nat_p
X6
∧
(
(
exactly2
X7
∧
(
(
¬
ordinal
X7
)
∧
(
¬
(
X6
∈
X4
)
)
)
)
∧
(
TransSet
X7
→
(
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
setsum_p
X8
)
)
∧
(
(
¬
exactly4
X6
)
∧
(
(
set_of_pairs
X7
∧
(
(
(
¬
(
∅
∈
X1
)
)
→
(
nat_p
X4
∧
(
¬
ordinal
X7
)
)
)
→
(
(
(
(
¬
equip
∅
X6
)
→
(
(
¬
exactly4
X7
)
∧
(
¬
atleast6
X6
)
)
)
→
(
(
¬
exactly2
X7
)
∧
(
¬
SNoLt
X6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
¬
exactly3
X7
)
)
)
)
∧
(
(
¬
atleast3
X0
)
∧
(
¬
exactly3
X5
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
(
(
¬
atleast6
X6
)
→
(
atleast3
X5
∧
(
(
(
(
¬
atleast5
X6
)
∧
(
(
¬
per_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
atleast2
X9
)
)
)
→
TransSet
X6
)
)
∧
(
ordinal
(
binunion
X7
∅
)
∧
(
¬
atleast4
(
SetAdjoin
X6
X7
)
)
)
)
→
(
(
¬
exactly5
X6
)
∧
(
¬
SNo
X7
)
)
)
)
→
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
atleast5
X0
)
→
(
Sing
X1
⊆
X7
)
)
→
(
(
(
¬
atleast3
X7
)
∧
(
atleast2
X7
→
(
(
(
¬
nat_p
X6
)
∧
(
(
¬
exactly5
X6
)
→
(
¬
atleast6
X5
)
)
)
∧
(
(
(
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
atleast5
X5
→
(
(
¬
atleast6
X8
)
∧
ordinal
X8
)
)
)
∧
(
exactly4
X9
→
exactly2
X8
)
)
∧
(
(
(
¬
(
X8
∈
X5
)
)
→
(
(
(
(
¬
PNo_upc
(
λX10 :
set
⇒
λX11 :
set
→
prop
⇒
(
(
X11
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
(
X11
X4
→
X11
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
→
(
¬
setsum_p
X4
)
→
X11
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
atleast3
X9
)
)
∧
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
X8
(
λX10 :
set
⇒
(
(
(
(
¬
exactly4
X10
)
→
exactly2
X10
→
(
atleast5
X10
∧
(
¬
atleastp
X10
(
proj0
∅
)
)
)
)
→
(
¬
atleast3
X8
)
→
SNo
X0
→
(
¬
SNo
X3
)
)
∧
(
(
(
exactly2
X9
→
(
ordinal
X10
∧
(
¬
atleast5
X0
)
)
)
→
(
(
X9
⊆
X3
)
∧
(
¬
atleastp
X10
X2
)
)
→
(
exactly4
X1
∧
(
atleast3
X9
→
(
¬
tuple_p
∅
X9
)
)
)
)
→
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
¬
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
¬
set_of_pairs
X2
)
)
)
→
(
¬
nat_p
X8
)
)
)
)
)
∧
atleast2
X6
)
→
(
atleast6
X7
∧
(
¬
atleast3
∅
)
)
→
(
¬
ordinal
X6
)
)
∧
nat_p
X0
)
)
)
)
→
(
(
atleast6
X7
∧
(
atleast5
X6
∧
(
(
(
TransSet
X3
∧
(
(
¬
exactly4
X7
)
→
(
(
(
(
¬
exactly2
X6
)
→
(
¬
exactly4
X7
)
)
∧
set_of_pairs
X7
)
∧
(
¬
setsum_p
X7
)
)
)
)
∧
atleast5
X1
)
∧
(
¬
atleast5
X2
)
)
)
)
∧
(
(
exactly3
X1
→
(
¬
exactly3
X7
)
)
→
(
equip
X6
X1
∧
(
(
(
¬
set_of_pairs
X6
)
→
(
¬
atleast3
X5
)
)
→
exactly5
X6
)
)
)
)
)
→
tuple_p
X7
X7
)
→
(
setsum_p
∅
∧
atleast4
X7
)
→
(
atleast6
X1
∧
(
set_of_pairs
∅
∧
(
(
¬
set_of_pairs
X1
)
→
exactly3
X3
→
nat_p
X6
)
)
)
)
→
(
¬
atleast3
X1
)
)
→
(
(
(
¬
set_of_pairs
X6
)
→
(
¬
nat_p
X7
)
)
∧
(
¬
atleast4
X6
)
)
→
(
(
¬
atleast6
X0
)
∧
nat_p
X7
)
)
→
tuple_p
X0
X0
)
→
(
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
¬
set_of_pairs
X5
)
→
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
¬
ordinal
∅
)
)
)
∧
(
exactly3
(
Inj1
X4
)
→
(
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
(
¬
setsum_p
X0
)
)
)
)
)
→
(
¬
partialorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
ordinal
X2
)
)
→
(
(
(
(
¬
nat_p
X4
)
∧
(
(
ordinal
X1
→
(
¬
atleast4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
X2
∈
X0
)
→
setsum_p
X5
→
(
(
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
atleast2
X5
)
)
∧
(
¬
atleast5
X5
)
)
)
∧
SNo
X3
)
)
)
∧
ordinal
X4
)
∧
(
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
(
exactly4
X5
∧
eqreln_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
stricttotalorder_i
(
λX8 :
set
⇒
λX9 :
set
⇒
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
(
(
(
¬
nat_p
X6
)
∧
(
(
(
(
(
¬
TransSet
∅
)
∧
(
¬
atleast2
X1
)
)
→
(
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
SNo
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
set_of_pairs
X7
)
)
∧
(
¬
ordinal
X7
)
)
→
(
¬
TransSet
∅
)
)
)
∧
(
¬
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X5
)
)
)
→
(
¬
atleast3
X1
)
)
∧
SNoLe
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X7
)
)
)
)
)
)
)
→
(
(
TransSet
X2
∧
(
(
ordinal
X2
→
(
¬
atleast6
∅
)
)
∧
(
(
(
¬
nat_p
X4
)
∧
(
(
atleast4
X0
→
(
¬
SNoLe
X5
X3
)
)
→
(
¬
exactly2
X2
)
)
)
∧
(
¬
ordinal
∅
)
)
)
)
∧
(
(
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
(
(
¬
ordinal
X0
)
→
(
¬
exactly2
X4
)
)
∧
(
(
¬
exactly2
∅
)
∧
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
(
(
¬
TransSet
X4
)
∧
(
¬
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
→
SNoLt
X1
X4
)
→
nat_p
X5
)
)
)
)
)
)
In Proofgold the corresponding term root is
16855b...
and proposition id is
71e7e1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMKaDL4suFdDiNMUBgGJn3mFnQCnCMwWYrg
)
∀X0 :
set
,
(
∀X1
⊆
∅
,
∀X2 :
set
,
(
¬
set_of_pairs
X0
)
→
(
∀X3
⊆
X1
,
∀X4 :
set
,
(
¬
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
exactly4
X8
∧
(
(
(
¬
atleastp
X5
X8
)
→
(
(
¬
exactly3
∅
)
∧
(
¬
ordinal
X8
)
)
)
→
nat_p
X7
→
(
¬
exactly2
X8
)
)
)
)
)
)
)
→
(
(
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
¬
atleast3
X3
)
→
TransSet
X1
→
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
∧
TransSet
X4
)
)
)
→
(
∃X1 :
set
,
(
(
¬
nat_p
X0
)
∧
(
(
∀X2
⊆
X1
,
(
∀X3
∈
X0
,
(
(
(
∀X4
∈
X2
,
(
(
¬
atleast3
X0
)
∧
(
(
¬
atleast2
X4
)
∧
(
(
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
inj
∅
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
λX5 :
set
⇒
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly5
X4
)
→
(
¬
(
X4
∈
X0
)
)
)
∧
(
∀X4 :
set
,
(
(
(
¬
exactly4
X4
)
→
(
¬
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
SNo
X2
)
)
∧
(
(
∀X4
∈
∅
,
(
¬
(
X4
∈
X0
)
)
)
→
(
∃X4 :
set
,
(
(
exactly2
X2
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
TransSet
X6
)
)
)
∧
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast4
X6
)
)
)
)
)
)
)
→
(
(
(
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∧
linear_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
exactly2
X1
∧
SNo
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
∧
(
(
∃X3 ∈
X0
,
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
atleast5
X1
)
)
)
∧
(
∀X3 :
set
,
(
(
∀X4 :
set
,
setsum_p
X1
)
→
(
(
∃X4 :
set
,
setsum_p
X0
)
∧
(
∀X4 :
set
,
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
→
(
exactly4
X3
∧
(
(
(
¬
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
exactly5
X2
∧
(
¬
SNo
(
Sing
X3
)
)
)
)
∧
(
(
¬
atleastp
X3
X2
)
∧
(
(
(
¬
PNo_downc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
¬
ordinal
X5
)
)
X4
(
λX5 :
set
⇒
totalorder_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
exactly5
X3
)
)
)
)
∧
setsum_p
X4
)
∧
(
(
¬
TransSet
X4
)
∧
(
exactly4
X0
→
(
(
atleast5
X4
∧
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
nat_p
(
Sep2
(
Inj1
X3
)
(
λX5 :
set
⇒
X4
)
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly4
X6
→
(
(
(
¬
atleast2
X6
)
→
(
SNo
(
proj0
X5
)
∧
(
(
(
¬
TransSet
X5
)
∧
ordinal
X5
)
∧
(
atleast2
X0
→
(
(
ordinal
X6
→
(
¬
atleast2
X6
)
)
→
(
¬
exactly2
X5
)
)
→
exactly3
X5
→
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
)
∧
(
¬
atleast5
X5
)
)
→
(
atleast6
∅
∧
(
(
¬
TransSet
X5
)
→
(
¬
exactly5
X0
)
)
)
→
(
¬
exactly2
∅
)
→
(
¬
exactly4
∅
)
)
)
∧
exactly2
X1
)
→
(
(
(
(
(
¬
exactly2
X0
)
→
exactly2
X3
)
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
exactly3
X0
)
)
∧
(
(
¬
exactly3
X1
)
∧
(
¬
exactly5
X4
)
)
)
→
exactly3
X4
)
)
∧
(
exactly3
X3
→
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
atleast2
X1
→
(
(
¬
TransSet
X0
)
∧
(
(
¬
equip
X3
X3
)
∧
(
(
TransSet
X0
→
(
(
¬
atleast5
X0
)
→
(
¬
atleast4
X1
)
)
→
(
¬
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast5
X6
)
)
)
→
(
¬
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly2
X6
)
)
)
)
)
→
(
¬
atleast6
X4
)
)
→
(
¬
atleast4
X4
)
)
∧
(
(
¬
SNoEq_
X1
X4
X1
)
∧
(
(
(
SNo
X4
∧
(
atleast2
X3
∧
(
(
¬
TransSet
X4
)
∧
(
¬
atleast2
X4
)
)
)
)
→
(
(
atleast4
X3
∧
(
(
(
(
inj
X3
∅
(
λX5 :
set
⇒
∅
)
→
atleast5
X1
)
→
(
¬
SNo
X3
)
)
∧
(
¬
exactly2
X4
)
)
→
(
(
(
¬
atleast5
X4
)
∧
(
¬
TransSet
X2
)
)
∧
exactly5
X4
)
)
)
∧
(
¬
(
X4
∈
X2
)
)
)
)
→
(
exactly3
X1
∧
(
exactly4
X1
→
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
∃X4 :
set
,
(
exactly3
X4
∧
(
atleast3
X4
∧
(
atleast4
X3
→
(
¬
atleast4
X2
)
)
)
)
)
)
)
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
174b7d...
and proposition id is
a8c472...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMW1vZr4QjbS5TdPp29oCGWVSd7xqdnfns1
)
∃X0 ∈
∅
,
∃X1 ∈
X0
,
(
(
∃X2 ∈
X1
,
∀X3
∈
X1
,
∀X4 :
set
,
(
(
(
¬
exactly2
X2
)
→
(
¬
exactly4
X2
)
)
∧
(
(
(
¬
exactly3
X4
)
→
nat_p
X2
)
→
exactly5
X2
)
)
)
∧
(
¬
reflexive_i
(
λX2 :
set
⇒
λX3 :
set
⇒
∃X4 :
set
,
(
(
¬
ordinal
X4
)
∧
(
(
¬
atleast3
X3
)
∧
(
(
(
exactly4
(
binunion
X2
X3
)
→
(
(
(
atleast4
X3
∧
(
(
¬
exactly3
(
V_
(
Inj0
X3
)
)
)
→
atleast5
X4
)
)
→
(
(
(
X0
=
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
TransSet
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
(
(
(
(
(
(
(
(
(
(
nat_p
X4
→
exactly2
X4
)
∧
(
(
¬
exactly3
X2
)
→
(
(
¬
atleast3
X3
)
∧
(
(
(
(
(
¬
atleast2
X3
)
∧
(
¬
nat_p
X2
)
)
→
(
¬
(
X3
∈
X3
)
)
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
linear_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast5
X7
)
)
)
)
∧
(
setsum_p
X4
→
atleast6
X3
)
)
)
)
)
→
(
¬
exactly2
X3
)
)
→
(
(
(
¬
atleast5
X0
)
∧
(
exactly5
∅
∧
(
¬
atleast3
X2
)
)
)
∧
(
¬
ordinal
X3
)
)
)
→
(
(
(
exactly4
X3
→
(
¬
exactly2
X4
)
)
∧
(
(
(
¬
exactly5
(
Inj0
X3
)
)
→
(
(
exactly2
X4
→
setsum_p
X4
)
→
(
TransSet
X1
∧
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
atleast5
X2
)
∧
(
(
¬
exactly3
X3
)
∧
(
atleast2
X4
∧
exactly4
∅
)
)
)
)
→
(
(
(
atleast2
(
UPair
X3
X2
)
∧
(
(
(
(
atleast2
X3
∧
atleast5
X1
)
∧
(
atleast2
X3
∧
(
¬
atleast6
X1
)
)
)
→
(
(
exactly5
X1
∧
(
atleast4
X4
→
ordinal
(
Inj1
X4
)
)
)
→
SNo
X2
→
(
atleast3
X4
∧
(
(
(
(
(
X1
⊆
X4
)
∧
(
¬
exactly2
X4
)
)
→
atleast5
X2
)
∧
(
¬
SNo
X1
)
)
∧
(
¬
ordinal
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
→
(
(
¬
nat_p
X1
)
∧
(
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
(
(
atleast6
X3
→
(
¬
TransSet
∅
)
)
∧
(
exactly3
X4
∧
atleast5
X3
)
)
)
)
→
(
(
(
¬
TransSet
X3
)
→
exactly2
X4
)
∧
(
exactly3
∅
→
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
(
¬
TransSet
X3
)
→
atleast4
X3
)
)
)
∧
(
¬
TransSet
X4
)
)
→
SNo_
X3
X4
)
→
(
exactly5
X4
∧
(
¬
atleast6
X4
)
)
)
→
atleast6
X1
)
→
(
(
¬
atleast2
X0
)
∧
(
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
nat_p
X1
)
)
)
→
exactly3
X3
)
→
(
SNo
∅
∧
(
¬
equip
X4
∅
)
)
)
→
atleast5
X3
)
∧
(
(
(
¬
exactly5
X2
)
→
(
equip
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
exactly4
X0
∧
(
(
(
¬
ordinal
X2
)
∧
(
¬
(
X2
=
X4
)
)
)
→
(
(
atleast2
(
setminus
X4
X3
)
→
exactly5
X4
→
exactly2
X4
→
atleast3
X4
)
∧
(
(
(
¬
atleast3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
atleast6
X4
)
)
∧
(
(
¬
set_of_pairs
X4
)
∧
(
¬
exactly3
X0
)
)
)
)
)
)
)
)
→
inj
∅
X1
(
λX5 :
set
⇒
X5
)
)
)
)
)
)
→
(
(
¬
atleast4
X2
)
∧
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
exactly4
X3
)
∧
(
(
(
inj
X4
X2
(
λX5 :
set
⇒
X5
)
∧
(
¬
nat_p
X4
)
)
→
atleast6
(
Inj0
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
(
¬
atleast6
X4
)
∧
(
exactly4
∅
∧
(
¬
exactly4
∅
)
)
)
)
)
∧
(
¬
ordinal
X0
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c5d9b2...
and proposition id is
050c4e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMVyNJFtYSim3YoDoPQXudCQL8zM9jqUPU9
)
∀X0
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
,
∃X1 ∈
∅
,
(
(
atleast2
X0
→
(
∀X2 :
set
,
nat_p
X2
→
(
∀X3
∈
X1
,
∀X4
⊆
X0
,
(
(
⋃
(
V_
X1
)
⊆
∅
)
→
(
exactly4
X3
∧
(
(
(
(
X0
⊆
X2
)
∧
(
(
¬
atleast5
X3
)
∧
(
(
¬
atleast5
X3
)
∧
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
nat_p
X4
∧
atleastp
X4
X2
)
→
set_of_pairs
X4
)
)
∧
(
(
(
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
¬
setsum_p
X4
)
)
∧
(
setsum_p
X3
→
(
(
¬
SNo
X4
)
∧
(
¬
(
∅
∈
X3
)
)
)
)
)
→
(
(
¬
setsum_p
X4
)
→
nat_p
X0
)
→
(
(
(
atleast2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
exactly4
X4
)
)
→
(
¬
atleast2
X4
)
)
∧
(
¬
nat_p
X4
)
)
)
)
)
)
→
(
(
¬
setsum_p
X1
)
∧
exactly2
X3
)
)
→
(
∀X3 :
set
,
(
∀X4 :
set
,
(
¬
SNo
X2
)
)
→
(
∃X4 ∈
X3
,
(
(
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
¬
atleast2
X4
)
)
∧
atleast5
X3
)
)
)
→
(
∃X3 :
set
,
(
(
X3
⊆
X1
)
∧
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
∧
(
∀X2
∈
X1
,
∀X3 :
set
,
(
(
¬
atleast3
X3
)
∧
atleast5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
ordinal
X2
)
)
In Proofgold the corresponding term root is
f13e79...
and proposition id is
b4aa11...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMJ59S5uzEhEZ4ZXEUTTBuEnrXu78g5Y6fW
)
∃X0 :
set
,
∀X1
⊆
X0
,
∃X2 :
set
,
(
(
X2
⊆
X0
)
∧
(
∀X3 :
set
,
(
∀X4 :
set
,
(
(
(
(
SNo
X4
→
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
setsum_p
X5
)
)
∧
(
(
(
(
(
¬
atleast6
(
⋃
X0
)
)
→
(
¬
exactly4
∅
)
)
→
setsum_p
(
If_i
(
(
(
(
¬
exactly3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
atleast5
X4
)
∧
atleast4
X3
)
∧
atleast2
X2
)
(
SetAdjoin
X0
∅
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
∧
(
atleast2
X3
→
(
exactly5
(
famunion
X2
(
λX5 :
set
⇒
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
exactly2
X4
)
)
)
)
∧
(
atleast4
X0
∧
(
¬
exactly2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
∧
(
atleast6
(
SNoElts_
X1
)
→
(
(
¬
atleast2
X4
)
∧
(
(
(
¬
atleast6
X2
)
→
atleast6
(
lam
X4
(
λX5 :
set
⇒
X2
)
)
→
(
(
(
(
¬
exactly4
X3
)
→
(
¬
atleast6
X2
)
)
→
(
(
(
(
(
¬
atleast3
X0
)
∧
(
(
(
(
¬
SNo
∅
)
∧
(
¬
(
X3
=
X2
)
)
)
∧
strictpartialorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
ordinal
X4
)
)
∧
(
(
(
(
exactly3
X4
∧
(
(
(
(
¬
atleast3
X4
)
∧
(
¬
exactly2
X4
)
)
→
(
(
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
(
¬
atleast6
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
¬
setsum_p
X0
)
→
exactly2
X3
)
∧
(
¬
SNo
X1
)
)
)
)
∧
(
¬
TransSet
X4
)
)
)
→
(
¬
ordinal
X3
)
)
)
∧
nat_p
(
Sing
(
Inj1
X3
)
)
)
→
(
atleast6
X3
∧
atleast4
X3
)
)
∧
(
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
(
¬
setsum_p
X4
)
∧
(
(
(
(
¬
exactly3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
(
¬
exactly2
X3
)
∧
(
¬
nat_p
X4
)
)
)
→
exactly2
X2
)
→
(
¬
exactly4
X2
)
→
(
(
¬
ordinal
∅
)
∧
(
¬
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
⊆
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
∧
atleast4
X3
)
∧
nat_p
X4
)
∧
SNo
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
∧
exactly3
X2
)
)
→
(
(
¬
exactly3
X0
)
→
(
¬
TransSet
X3
)
)
→
(
(
(
SNo
X4
→
atleast2
X3
)
→
(
¬
atleast6
X1
)
)
→
(
(
atleast4
X0
∧
(
¬
SNo
X4
)
)
∧
(
¬
linear_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast6
X5
)
)
)
)
)
→
(
¬
atleast2
X4
)
)
)
→
(
(
¬
TransSet
X2
)
∧
SNo
X2
)
)
)
→
(
¬
exactly2
X4
)
)
→
(
X2
∈
∅
)
→
(
atleast5
X4
∧
(
(
(
¬
exactly4
X3
)
∧
(
¬
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
nat_p
X1
)
)
)
→
(
exactly3
X2
∧
setsum_p
X2
)
)
)
In Proofgold the corresponding term root is
eed21d...
and proposition id is
7644b9...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMS3EDbBZFJzgMToBqJSBQHfXgffL8sHP8c
)
∀X0
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∀X1 :
set
,
(
(
(
∀X2 :
set
,
(
(
equip
X0
X0
∧
(
(
¬
exactly5
X0
)
→
(
∀X3 :
set
,
∀X4
⊆
X0
,
(
(
(
¬
atleast6
X3
)
∧
(
¬
exactly4
X2
)
)
∧
atleast4
X1
)
)
)
)
∧
(
∃X3 :
set
,
(
(
¬
exactly4
X3
)
∧
(
¬
atleast4
X3
)
)
)
)
→
(
∃X3 ∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
(
¬
atleast6
X0
)
)
)
→
(
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
¬
exactly2
X0
)
)
)
∧
(
∀X2
∈
X0
,
(
∀X3
∈
X2
,
∀X4
⊆
∅
,
nat_p
X4
→
set_of_pairs
X3
→
ordinal
X2
)
→
(
∀X3 :
set
,
atleast6
X2
→
(
∃X4 :
set
,
atleast5
X4
)
→
(
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
ordinal
∅
)
)
)
)
)
)
)
∧
atleast5
X1
)
→
(
¬
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
In Proofgold the corresponding term root is
f04de3...
and proposition id is
3b2cfe...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMdWUezaeWNVY2Frq8i7d1NBK5g4kg5Boh9
)
∀X0
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
∃X1 :
set
,
(
(
∀X2 :
set
,
(
∃X3 :
set
,
(
(
atleast2
X2
→
(
¬
atleast3
X3
)
)
∧
(
∀X4 :
set
,
(
atleast3
X4
→
(
¬
atleast4
X3
)
→
(
(
(
(
¬
exactly3
∅
)
→
(
(
(
(
¬
SNoLt
X3
X3
)
∧
(
(
(
(
¬
exactly5
X3
)
→
TransSet
X3
)
∧
(
¬
atleast6
∅
)
)
∧
(
(
(
(
(
¬
exactly2
X4
)
∧
(
(
(
(
(
(
¬
atleast4
(
⋃
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
exactly2
X2
→
ordinal
X4
)
)
∧
(
(
(
(
atleast2
X1
→
(
¬
exactly3
X3
)
→
(
(
¬
atleast5
X3
)
→
atleast2
X4
)
→
nat_p
X4
)
∧
(
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
=
∅
)
)
)
)
→
(
(
¬
exactly3
(
Sing
∅
)
)
→
(
¬
exactly4
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
¬
atleast4
X3
)
→
(
(
(
¬
TransSet
X2
)
→
(
¬
atleast3
X3
)
)
∧
(
(
¬
ordinal
∅
)
∧
(
¬
PNo_upc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
(
¬
X6
X3
)
∧
nat_p
X4
)
)
X0
(
λX5 :
set
⇒
(
¬
exactly3
X2
)
)
)
)
)
)
)
∧
(
(
(
nat_p
X3
∧
(
¬
atleast2
X4
)
)
∧
(
(
¬
ordinal
X2
)
→
(
¬
irreflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
TransSet
(
Sep2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
(
λX7 :
set
⇒
X6
)
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
exactly3
X0
→
tuple_p
X3
X6
)
∧
exactly2
X2
)
→
atleast2
X7
)
)
)
∧
exactly5
X5
)
)
)
)
)
∧
(
(
(
atleast2
X3
∧
(
TransSet
X4
∧
exactly3
X2
)
)
∧
(
PNoLt
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
(
λX5 :
set
⇒
(
(
¬
exactly1of2
(
atleast6
∅
∧
atleastp
X4
X5
)
(
(
¬
atleastp
X5
(
ordsucc
X4
)
)
∧
(
¬
exactly2
X5
)
)
)
∧
(
(
(
(
¬
exactly3
X4
)
∧
nat_p
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
exactly4
X5
)
)
→
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
X4
(
λX5 :
set
⇒
(
¬
TransSet
X5
)
)
∧
atleast3
X3
)
)
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
atleast3
X0
∧
(
¬
binop_on
X6
(
λX7 :
set
⇒
λX8 :
set
⇒
∅
)
)
)
∧
(
(
¬
SNo_
X0
X5
)
→
(
¬
exactly5
X3
)
)
)
)
∧
(
¬
ordinal
X3
)
)
→
atleast6
X0
)
)
)
)
∧
(
(
(
(
¬
exactly1of3
(
(
(
¬
atleast2
X4
)
→
(
atleast2
X0
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
nat_p
X6
)
)
→
(
atleast5
X0
→
exactly2
X4
→
TransSet
X3
→
(
exactly3
X4
∧
(
X3
∈
X0
)
)
)
→
(
atleast4
X4
∧
(
(
(
¬
set_of_pairs
X3
)
∧
(
¬
atleast6
X1
)
)
∧
(
atleast2
X3
→
(
(
¬
setsum_p
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
exactly5
X3
)
→
(
(
exactly5
X4
→
(
¬
SNo
X3
)
)
∧
(
¬
atleast4
X2
)
)
)
)
)
)
∧
(
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
atleast6
X1
)
)
)
)
∧
(
(
(
¬
exactly5
X2
)
∧
(
¬
ordinal
X4
)
)
→
(
(
(
(
nat_p
X2
→
(
exactly2
X2
∧
(
(
(
(
(
¬
exactly3
X3
)
∧
(
(
¬
exactly3
X0
)
∧
(
(
SNo
X4
∧
exactly3
X4
)
→
(
(
atleast6
∅
→
(
(
(
(
exactly5
X3
∧
setsum_p
X4
)
→
(
(
exactly2
X2
→
exactly3
X4
)
∧
(
¬
exactly3
∅
)
)
)
→
exactly2
X4
→
(
(
¬
set_of_pairs
(
SNoElts_
X3
)
)
∧
(
(
¬
exactly2
X3
)
∧
ordinal
X2
)
)
)
∧
TransSet
X4
)
)
∧
PNoLt_
X2
(
λX5 :
set
⇒
(
¬
SNo
∅
)
)
(
λX5 :
set
⇒
(
(
¬
atleast3
X4
)
∧
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
→
PNo_upc
(
λX5 :
set
⇒
λX6 :
set
→
prop
⇒
(
(
¬
X6
X4
)
→
X6
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
atleast6
X8
)
)
)
)
X4
(
λX5 :
set
⇒
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
∧
SNo
(
ordsucc
∅
)
)
∧
(
(
¬
nat_p
X3
)
→
(
¬
atleast3
X0
)
→
(
¬
setsum_p
X4
)
)
)
∧
(
nat_p
X4
∧
PNoLt_
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
(
λX5 :
set
⇒
exactly4
∅
)
(
λX5 :
set
⇒
atleast6
X4
→
(
(
atleast3
X4
∧
TransSet
X3
)
∧
(
(
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
(
(
(
¬
ordinal
X5
)
→
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
setsum_p
X5
)
∧
(
atleast5
X5
∧
SNoLe
X5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
→
(
(
¬
atleast4
(
Inj1
∅
)
)
∧
(
¬
atleast4
X2
)
)
→
(
¬
atleast5
X5
)
)
)
)
)
)
)
)
→
(
¬
atleast5
X2
)
)
∧
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
¬
SNoEq_
X6
X5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
∧
(
¬
tuple_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
X5
)
)
∧
atleast6
X6
)
)
→
(
atleast6
X6
∧
(
¬
atleast5
X5
)
)
)
→
(
¬
atleast6
X5
)
)
)
)
∧
(
(
¬
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
exactly3
X0
→
(
(
X6
=
X5
)
∧
(
(
(
¬
(
X0
⊆
X5
)
)
∧
(
(
(
(
¬
exactly4
X5
)
→
exactly2
X5
)
∧
(
atleast5
X6
→
(
(
(
(
(
nat_p
X0
∧
(
(
¬
atleast3
X6
)
→
atleast2
X6
→
exactly4
X5
)
)
∧
(
atleast5
X6
∧
atleast5
X5
)
)
∧
(
(
¬
atleast6
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
exactly2
X5
)
)
∧
(
(
¬
exactly2
X2
)
→
(
(
TransSet
X6
∧
(
(
(
¬
exactly2
X0
)
→
ordinal
X5
)
∧
(
¬
atleast3
X4
)
)
)
∧
set_of_pairs
(
Sing
X6
)
)
)
)
∧
(
¬
exactly2
X6
)
)
)
)
→
(
¬
set_of_pairs
X4
)
)
)
→
(
¬
(
X6
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
(
¬
atleast2
X5
)
∧
nat_p
X6
)
→
(
atleast6
X5
∧
(
(
¬
exactly4
X0
)
∧
exactly3
X6
)
)
→
(
(
¬
TransSet
X0
)
∧
(
(
(
¬
ordinal
X6
)
∧
(
equip
X6
X0
→
(
atleast4
X6
∧
(
¬
exactly5
X5
)
)
→
(
(
¬
atleast2
X1
)
∧
(
TransSet
X6
∧
(
(
¬
atleastp
X5
X3
)
∧
TransSet
X6
)
)
)
→
TransSet
X0
)
)
→
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
∧
(
tuple_p
X4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∧
(
¬
nat_p
X0
)
)
)
)
)
)
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast2
X0
→
(
(
atleast4
∅
∧
(
¬
SNoLt
X6
X6
)
)
∧
(
(
(
(
¬
atleast2
(
⋃
X6
)
)
→
(
¬
TransSet
∅
)
)
→
(
¬
ordinal
X5
)
)
→
(
(
(
¬
set_of_pairs
(
⋃
X5
)
)
→
(
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
exactly4
X7
→
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
ordinal
∅
)
→
exactly4
X5
)
∧
(
atleast2
X0
→
(
¬
atleast3
X5
)
→
(
¬
TransSet
X1
)
)
)
)
→
(
symmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
ordinal
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∧
atleast4
(
SNoLev
∅
)
)
)
∧
symmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
SNoLe
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
exactly4
(
ReplSep
X4
(
λX7 :
set
⇒
(
(
(
(
(
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
(
(
(
(
(
¬
ordinal
X8
)
→
(
(
atleast3
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
(
¬
setsum_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
(
¬
tuple_p
X9
X5
)
∧
(
exactly4
X0
→
atleast5
X8
)
)
)
∧
(
(
(
exactly3
X3
∧
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
nat_p
X9
)
∧
(
¬
exactly3
X8
)
)
)
)
∧
(
reflexive_i
(
λX10 :
set
⇒
λX11 :
set
⇒
(
¬
atleast6
X11
)
→
atleast5
X11
)
∧
(
(
exactly4
X8
∧
(
(
¬
atleast6
X9
)
∧
(
¬
exactly3
X8
)
)
)
→
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
)
→
(
¬
exactly2
∅
)
)
→
(
¬
exactly4
X9
)
)
∧
(
TransSet
∅
→
exactly2
X7
)
)
→
(
¬
atleast2
(
binintersect
X9
(
ReplSep
X8
(
λX10 :
set
⇒
(
(
¬
atleast3
X10
)
∧
TransSet
X9
)
)
(
λX10 :
set
⇒
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
→
atleast6
X9
)
)
∧
(
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
exactly3
(
UPair
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
exactly5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
(
(
¬
atleast3
X7
)
∧
(
(
¬
SNo
(
𝒫
X6
)
)
∧
inj
X6
X7
(
λX8 :
set
⇒
X0
)
)
)
→
(
(
¬
nat_p
X7
)
∧
exactly4
(
UPair
X7
X6
)
)
)
∧
atleast2
X7
)
)
)
∧
(
¬
atleast5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
¬
nat_p
X7
)
)
→
exactly3
∅
→
atleast6
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
(
λX7 :
set
⇒
X6
)
)
→
ordinal
X5
)
→
(
X5
∈
X5
)
)
)
)
)
(
(
¬
nat_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
exactly3
X5
)
∧
PNoLe
X0
(
λX7 :
set
⇒
(
¬
exactly5
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
X5
(
λX7 :
set
⇒
(
(
(
¬
TransSet
X6
)
→
(
(
(
(
(
¬
atleast3
(
Inj1
X3
)
)
→
(
setsum_p
X6
∧
(
atleast6
X6
→
SNoLe
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
(
(
(
(
(
¬
(
X4
∈
X1
)
)
∧
(
¬
setsum_p
X4
)
)
∧
atleast4
X6
)
→
(
¬
atleast3
X7
)
)
∧
(
exactly5
X7
→
(
X7
∈
∅
)
)
)
)
∧
(
totalorder_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
setsum_p
X9
)
)
∧
(
¬
setsum_p
X6
)
)
)
∧
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
¬
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
∧
(
¬
ordinal
X7
)
)
→
(
¬
exactly3
X0
)
)
)
)
∧
(
(
(
¬
set_of_pairs
(
famunion
X6
(
λX8 :
set
⇒
X1
)
)
)
→
(
(
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
∈
X7
)
∧
(
(
¬
inj
∅
X4
(
λX8 :
set
⇒
X7
)
)
→
(
¬
bij
X7
X2
(
λX8 :
set
⇒
X5
)
)
)
)
→
(
¬
(
X6
∈
X2
)
)
)
→
nat_p
X6
)
→
(
¬
atleast3
X6
)
→
(
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
exactly4
X0
)
→
(
¬
atleast3
X1
)
)
)
)
)
→
atleast6
X6
)
→
(
(
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
stricttotalorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
¬
exactly5
X7
)
∧
(
(
(
¬
atleast6
(
Sing
(
Inj1
∅
)
)
)
→
(
¬
exactly3
X8
)
→
(
(
¬
nat_p
X7
)
∧
atleast6
X8
)
)
→
atleast6
X8
)
)
)
)
)
→
(
(
(
ordinal
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
(
(
exactly3
X3
→
(
atleast2
X0
∧
TransSet
X4
)
)
→
(
¬
SNo
X3
)
)
∧
set_of_pairs
X3
)
)
∧
exactly5
X3
)
∧
(
atleast3
X4
∧
(
(
¬
atleast6
X4
)
∧
(
(
¬
setsum_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
atleast4
X3
∧
equip
(
Unj
(
V_
X4
)
)
X3
)
)
)
)
)
)
∧
(
(
(
exactly2
X3
→
(
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
¬
setsum_p
X4
)
)
)
→
(
¬
exactly5
∅
)
)
→
(
(
(
atleast3
∅
→
(
(
(
¬
nat_p
X3
)
∧
nat_p
X3
)
∧
(
¬
PNoLt_
∅
(
λX5 :
set
⇒
ordinal
X5
)
(
λX5 :
set
⇒
(
¬
set_of_pairs
X5
)
→
(
¬
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
→
(
atleast4
X4
→
(
¬
ordinal
∅
)
)
→
(
atleast6
X4
→
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
¬
exactly2
X4
)
)
∧
exactly2
X3
)
)
)
→
(
¬
ordinal
X4
)
)
→
(
trichotomous_or_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
ordinal
X0
)
)
∧
(
¬
exactly3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
∧
exactly4
X4
)
→
(
¬
SNo_
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X2
)
)
∧
(
TransSet
X1
∧
(
¬
set_of_pairs
X3
)
)
)
)
)
→
(
(
(
¬
ordinal
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
equip
(
Sing
X4
)
X4
∧
(
¬
exactly4
X3
)
)
)
→
(
¬
setsum_p
X0
)
)
→
(
¬
ordinal
X4
)
)
→
(
¬
atleast3
X4
)
)
→
(
atleast3
X4
∧
(
(
(
¬
exactly4
X2
)
∧
(
¬
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
∈
X4
)
)
)
→
(
¬
SNo
X3
)
)
)
)
)
∧
exactly4
(
𝒫
X2
)
)
→
(
(
¬
atleast6
X3
)
→
atleast4
∅
)
→
(
(
(
(
(
¬
atleast3
X3
)
→
(
(
¬
exactly5
X0
)
∧
(
(
(
atleast2
X0
∧
(
¬
setsum_p
X4
)
)
→
(
¬
exactly5
X2
)
)
∧
(
(
¬
nat_p
X2
)
∧
(
(
atleast5
X1
∧
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
atleast6
X6
→
nat_p
X1
)
∧
(
(
(
(
(
(
¬
nat_p
X6
)
∧
(
(
(
atleast2
X6
→
(
(
nat_p
(
⋃
X6
)
∧
(
ordinal
X5
∧
(
(
(
(
(
(
¬
exactly2
X5
)
→
(
(
¬
atleast2
X5
)
∧
(
set_of_pairs
X0
∧
(
¬
nat_p
X3
)
)
)
)
→
(
(
SNo
X1
→
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
X6
∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
¬
atleast2
X2
)
)
→
exactly2
X6
)
∧
ordinal
X4
)
)
)
→
(
(
(
(
(
(
TransSet
X6
∧
(
atleast2
X0
∧
(
(
X6
∈
X6
)
→
(
¬
exactly5
X0
)
)
)
)
∧
TransSet
X3
)
→
(
¬
atleast2
X0
)
)
→
(
(
¬
ordinal
X5
)
∧
SNo
X6
)
)
∧
(
¬
(
∅
∈
X6
)
)
)
∧
(
¬
SNo_
X5
X5
)
)
)
→
(
¬
setsum_p
X2
)
→
TransSet
(
PSNo
X5
(
λX7 :
set
⇒
(
¬
exactly2
X0
)
)
)
)
→
(
¬
atleast3
X6
)
)
→
(
(
exactly4
X5
∧
(
(
(
(
¬
atleast2
X5
)
→
(
¬
atleast4
X5
)
)
∧
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
atleast2
X6
)
)
)
∧
(
(
setsum_p
X6
∧
(
(
(
(
¬
SNo_
X5
X6
)
→
(
¬
atleast5
X5
)
)
→
(
atleast5
X6
∧
(
(
exactly5
X5
→
linear_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
(
ordinal
X8
→
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
→
(
atleast6
X8
∧
(
(
(
(
¬
atleast4
X7
)
→
(
¬
TransSet
X8
)
)
∧
SNoEq_
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
X7
X7
)
∧
(
¬
setsum_p
X2
)
)
)
→
(
(
(
exactly4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
(
¬
SNo_
X7
∅
)
)
→
exactly5
∅
)
∧
(
(
¬
atleast5
X1
)
→
atleast6
X7
→
reflexive_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
(
exactly4
(
𝒫
X9
)
∧
(
atleast2
X9
∧
(
(
¬
TransSet
X0
)
∧
SNoLt
∅
X9
)
)
)
∧
(
(
(
(
(
¬
ordinal
X10
)
→
(
X10
∈
X10
)
→
(
(
(
¬
equip
X9
X9
)
∧
exactly4
X9
)
∧
(
¬
PNoLe
X9
(
λX11 :
set
⇒
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
X10
(
λX11 :
set
⇒
TransSet
X11
)
)
)
)
∧
(
atleast2
X10
∧
(
atleast6
X10
→
(
¬
atleast6
X9
)
)
)
)
→
(
¬
atleast4
X0
)
)
∧
atleast3
X9
)
)
)
)
)
)
∧
(
(
¬
linear_i
(
λX9 :
set
⇒
λX10 :
set
⇒
(
¬
atleast2
(
SNoLev
X0
)
)
)
)
∧
(
(
(
(
(
(
¬
atleast3
X6
)
→
exactly4
X1
)
∧
(
¬
exactly3
X7
)
)
∧
atleast2
X6
)
→
atleast5
X4
)
→
exactly3
∅
)
)
)
)
)
→
(
¬
TransSet
X5
)
)
)
)
→
SNo
X6
)
)
→
exactly5
X6
→
strictpartialorder_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
TransSet
X8
∧
(
¬
atleast3
X7
)
)
∧
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
)
→
(
¬
atleast3
X5
)
)
)
→
(
(
¬
exactly2
(
binunion
∅
X6
)
)
→
(
¬
exactly2
X4
)
)
→
ordinal
X6
→
(
(
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast5
X6
)
∧
(
¬
set_of_pairs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
atleast3
X4
)
)
∧
ordinal
X2
)
→
(
¬
ordinal
X2
)
)
)
)
∧
(
ordinal
X2
∧
(
¬
nat_p
X4
)
)
)
)
→
(
(
¬
atleast3
∅
)
→
(
¬
exactly3
X4
)
)
→
atleast4
X3
)
)
)
)
→
(
¬
atleast3
(
proj0
X4
)
)
→
(
(
¬
(
X3
∈
X0
)
)
∧
(
(
¬
exactly4
X4
)
→
(
¬
ordinal
X3
)
)
)
→
exactly5
X2
)
∧
(
¬
atleast2
X4
)
)
∧
(
(
¬
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
(
(
(
¬
SNo
X2
)
∧
exactly3
X3
)
∧
(
(
atleast4
(
SNoElts_
X0
)
→
(
(
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
→
atleast2
X3
)
∧
(
(
(
¬
SNo
X3
)
→
(
atleast6
X3
∧
(
¬
setsum_p
X4
)
)
)
→
(
¬
exactly5
X4
)
)
)
)
∧
(
(
X3
∈
X4
)
→
exactly2
X2
)
)
)
→
(
(
(
¬
atleast6
X2
)
∧
(
TransSet
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
→
set_of_pairs
X3
)
)
∧
(
(
¬
exactly4
X4
)
∧
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
(
(
(
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
exactly4
X7
∧
atleast3
X7
)
)
∧
exactly5
X6
)
∧
(
(
(
exactly4
X5
∧
(
(
(
(
¬
TransSet
X0
)
∧
(
(
(
¬
atleast5
X0
)
∧
(
¬
atleast3
X0
)
)
→
(
¬
exactly1of2
(
ordinal
X5
∧
(
(
¬
ordinal
∅
)
∧
(
exactly5
∅
→
(
X4
∈
X6
)
)
)
)
(
bij
X6
X5
(
λX7 :
set
⇒
X7
)
)
)
)
)
∧
(
¬
atleast3
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
(
¬
exactly5
X6
)
∧
(
¬
tuple_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
X0
)
)
)
)
→
(
(
¬
exactly1of3
(
(
(
(
¬
nat_p
X5
)
∧
(
¬
ordinal
X6
)
)
→
(
¬
exactly4
∅
)
)
→
(
equip
X5
X6
∧
exactly2
X3
)
)
(
(
(
(
(
¬
nat_p
X4
)
→
(
¬
TransSet
(
Inj0
X5
)
)
)
→
(
(
¬
atleast6
X4
)
∧
exactly5
X3
)
)
→
(
nat_p
X6
∧
(
(
¬
set_of_pairs
X6
)
→
SNo
X1
)
)
)
∧
TransSet
X2
)
(
ordinal
X5
)
)
∧
(
¬
ordinal
X0
)
)
)
∧
(
¬
exactly3
X1
)
)
)
∧
(
(
¬
exactly2
X5
)
∧
TransSet
X5
)
)
∧
(
(
¬
exactly4
X2
)
∧
atleast4
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
exactly3
X6
)
→
atleast3
X3
)
)
)
)
)
)
∧
(
(
¬
atleast5
X3
)
→
(
(
¬
atleast5
X4
)
∧
(
ordinal
X3
∧
(
(
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
→
(
(
¬
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
exactly5
X5
→
exactly2
X5
→
(
¬
equip
X6
X5
)
)
→
equip
(
𝒫
X5
)
X0
)
)
∧
(
¬
atleast3
X3
)
)
)
→
(
¬
atleast3
X2
)
)
)
)
)
)
)
→
(
¬
exactly3
X3
)
)
)
)
∧
setsum_p
∅
)
∧
(
¬
atleast5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
∧
(
(
¬
exactly4
X3
)
∧
(
atleast6
X4
→
(
(
¬
exactly5
∅
)
∧
(
atleast6
X4
→
(
(
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
→
(
¬
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
¬
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
∈
X0
)
)
)
→
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
∧
(
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
setsum_p
X3
)
)
)
→
SNo_
X4
∅
)
)
)
→
(
∃X3 ∈
X0
,
∃X4 :
set
,
(
¬
exactly5
X3
)
→
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
∃X2 ∈
X1
,
(
¬
exactly4
(
Sing
X0
)
)
)
)
In Proofgold the corresponding term root is
b6cd8a...
and proposition id is
0cde7e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMxFuMZ8p3wtyxeQnwJuNFCmsPxdW11TNG
)
∀X0 :
set
,
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
∀X4
⊆
X3
,
exactly4
X3
)
)
)
∧
(
atleast5
X1
→
(
∃X3 :
set
,
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
nat_p
X3
)
)
)
)
∧
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
¬
TransSet
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
In Proofgold the corresponding term root is
3d7792...
and proposition id is
69504b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMNrc2KCLxTd3yiUGAs56oZUEsPV1EYTR6Y
)
∃X0 :
set
,
∀X1
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
,
∀X2 :
set
,
(
(
∃X3 :
set
,
(
(
X3
⊆
Unj
∅
)
∧
(
∃X4 ∈
X0
,
exactly3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
nat_p
(
Unj
X3
)
)
)
)
)
∧
(
∀X3
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
(
(
(
∃X4 :
set
,
(
(
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
∧
set_of_pairs
X0
)
∧
(
(
(
¬
exactly3
X3
)
∧
(
atleast4
(
𝒫
X2
)
∧
(
(
¬
exactly2
X2
)
→
(
¬
atleast3
X1
)
)
)
)
∧
(
(
¬
atleast5
X2
)
→
atleast4
X3
)
)
)
)
∧
(
∀X4
⊆
combine_funcs
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
X0
(
λX5 :
set
⇒
X1
)
(
λX5 :
set
⇒
X0
)
X3
,
atleastp
∅
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
∃X4 :
set
,
(
atleast5
X3
∧
nat_p
∅
)
)
)
)
)
In Proofgold the corresponding term root is
cdd3e5...
and proposition id is
5b34db...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMRX9PjGutHoE4dx51GKvGBQ3Tb5QadGwhM
)
∀X0 :
set
,
(
∃X1 :
set
,
(
(
¬
atleast4
X0
)
∧
(
nat_p
X1
∧
(
∃X2 ∈
X0
,
(
(
(
(
¬
exactly2
∅
)
→
(
∃X3 :
set
,
(
(
(
∃X4 ∈
X2
,
(
¬
tuple_p
X2
X4
)
)
∧
(
∀X4 :
set
,
(
¬
exactly4
X4
)
→
(
(
(
¬
exactly2
X3
)
∧
(
(
¬
exactly4
X4
)
∧
(
(
¬
exactly4
(
If_i
(
(
¬
atleast2
X3
)
∧
(
(
¬
atleast6
X3
)
→
(
SNo
X2
→
(
¬
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
→
(
(
¬
atleast6
X3
)
∧
SNo_
X2
X3
)
)
)
X3
X3
)
)
→
(
TransSet
X1
∧
SNo
X4
)
→
(
(
(
¬
exactly2
X1
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast5
∅
)
)
∧
(
(
¬
atleast6
X4
)
→
(
¬
exactly4
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
∧
exactly5
X3
)
)
)
∧
(
∃X4 :
set
,
(
(
X4
⊆
X0
)
∧
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
)
)
)
∧
(
∀X3 :
set
,
∃X4 :
set
,
(
(
¬
atleast5
X3
)
∧
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
∧
(
∃X3 :
set
,
setsum_p
X1
)
)
)
)
)
)
→
(
∀X1 :
set
,
exactly2
X1
)
In Proofgold the corresponding term root is
33bcc0...
and proposition id is
624717...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcpceQpQNqYrFSdTcmp9eocBpjej63CHBi
)
∀X0
⊆
setprod
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
∀X1
⊆
∅
,
∀X2
⊆
⋃
X0
,
(
(
∀X3
∈
X1
,
∀X4 :
set
,
(
(
PNoLe
X3
(
λX5 :
set
⇒
(
(
(
¬
atleast5
X4
)
∧
(
set_of_pairs
X1
∧
(
(
¬
exactly4
(
V_
∅
)
)
∧
(
(
atleast4
X1
∧
exactly2
X4
)
∧
(
¬
exactly4
X2
)
)
)
)
)
∧
(
¬
atleast4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
)
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
λX5 :
set
⇒
exactly5
∅
)
→
(
¬
nat_p
X2
)
)
∧
atleast5
X3
)
)
∧
(
ordinal
∅
→
ordinal
(
SetAdjoin
X1
X0
)
)
)
In Proofgold the corresponding term root is
295a38...
and proposition id is
88d416...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMPncFFKDBfyAiv3xiPJaCwBjeHmqB2fhpz
)
∃X0 :
set
,
∀X1 :
set
,
(
∀X2 :
set
,
(
∀X3
∈
X2
,
(
¬
ordinal
X2
)
)
→
(
∃X3 :
set
,
(
exactly2
X2
∧
(
(
(
(
∀X4 :
set
,
(
exactly4
X3
∧
(
(
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
∧
(
(
(
atleast6
X2
→
(
¬
nat_p
X2
)
)
∧
ordinal
X2
)
∧
(
(
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
atleast6
X2
)
)
∧
(
(
¬
exactly4
∅
)
→
(
(
(
¬
atleast2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
(
(
¬
exactly3
∅
)
∧
(
(
¬
atleast2
X4
)
∧
(
¬
exactly4
X1
)
)
)
→
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
)
)
)
→
(
¬
exactly2
X3
)
)
)
)
→
(
¬
atleast2
∅
)
)
→
(
∀X4 :
set
,
(
¬
atleast3
∅
)
→
(
(
¬
TransSet
X3
)
→
exactly5
X3
→
setsum_p
(
In_rec_i
(
λX5 :
set
⇒
λX6 :
set
→
set
⇒
X5
)
X3
)
)
→
exactly2
X4
→
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
∧
(
∀X4 :
set
,
(
¬
(
X4
∈
X2
)
)
)
)
)
)
)
→
(
∀X2 :
set
,
atleast6
X1
→
(
(
exactly5
X0
→
(
∃X3 :
set
,
(
(
X3
⊆
X2
)
∧
(
¬
(
X3
⊆
X3
)
)
)
)
)
∧
atleast3
X1
)
)
In Proofgold the corresponding term root is
9506b0...
and proposition id is
fb4353...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMKGg6bjyafH3UEZ9AVdDgdKtDs2o3L2K5M
)
∃X0 :
set
,
(
(
X0
⊆
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
∃X1 :
set
,
(
(
∃X2 :
set
,
(
(
∃X3 ∈
X1
,
(
¬
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
∃X3 :
set
,
(
(
∀X4
∈
X3
,
(
(
¬
nat_p
X3
)
∧
(
¬
atleast5
X0
)
)
)
∧
(
∀X4
⊆
ordsucc
X3
,
(
(
¬
TransSet
(
ordsucc
X4
)
)
∧
(
(
(
(
(
¬
atleast2
X4
)
→
atleast2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast3
(
Inj0
X2
)
)
∧
(
¬
SNo
X4
)
)
→
exactly2
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
→
(
set_of_pairs
∅
→
(
¬
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast2
X0
)
)
)
→
(
(
(
atleast4
X0
→
(
X4
∈
X4
)
)
→
(
(
(
X1
∈
X3
)
→
ordinal
X3
)
∧
(
¬
atleast4
X3
)
)
)
→
(
(
(
(
(
¬
exactly4
(
Inj1
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
exactly3
∅
)
→
ordinal
X3
)
→
atleast5
(
𝒫
X4
)
)
→
(
(
nat_p
∅
∧
SNo
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
(
(
nat_p
X4
∧
(
¬
exactly5
X1
)
)
→
SNo_
X3
X2
)
→
exactly4
X2
)
→
exactly5
X2
→
(
TransSet
X3
∧
(
exactly2
X0
∧
(
¬
exactly3
X4
)
)
)
)
→
exactly4
X3
)
→
(
(
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
TransSet
X6
)
∧
(
(
¬
TransSet
X4
)
∧
(
atleast2
X2
∧
exactly2
X4
)
)
)
∧
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
)
)
)
)
)
)
∧
(
(
∀X2
∈
X1
,
atleast2
X2
)
∧
(
exactly3
X1
∧
exactly3
X1
)
)
)
)
)
In Proofgold the corresponding term root is
76964b...
and proposition id is
e0d128...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMHHNhV8Kf4qQtiRNYhHucq3HF6gAE2ZmxL
)
∀X0 :
set
,
∀X1 :
set
,
(
(
∃X2 ∈
X0
,
∀X3 :
set
,
∃X4 :
set
,
(
¬
setsum_p
X0
)
)
→
(
∀X2
⊆
∅
,
(
(
(
∃X3 :
set
,
(
¬
atleast5
X3
)
)
→
(
∃X3 :
set
,
(
(
X3
⊆
𝒫
X1
)
∧
(
(
∃X4 ∈
X3
,
(
(
totalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
TransSet
X6
)
)
→
TransSet
X3
)
∧
(
(
¬
ordinal
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
exactly2
∅
)
)
)
→
(
¬
set_of_pairs
X2
)
)
)
)
)
∧
(
¬
atleast4
X1
)
)
)
)
→
(
¬
SNo
X1
)
In Proofgold the corresponding term root is
5e3f17...
and proposition id is
5a01c5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMJqB1iZtcmcnmmj4zHkt6tvjHWaRRjgcAf
)
∃X0 :
set
,
(
(
X0
⊆
Sing
(
Inj1
∅
)
)
∧
(
∃X1 :
set
,
(
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
∧
(
∀X2
∈
X1
,
(
(
∃X3 ∈
X2
,
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
¬
TransSet
X4
)
)
)
∧
(
∀X3 :
set
,
∀X4 :
set
,
(
¬
TransSet
(
Sing
X3
)
)
→
(
(
¬
exactly3
X3
)
∧
(
(
¬
exactly5
X3
)
→
(
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
(
(
(
¬
PNoLt
X1
(
λX5 :
set
⇒
SNoLt
X4
X1
→
atleast2
X1
)
X4
(
λX5 :
set
⇒
(
(
SNo_
X4
X5
∧
(
(
(
(
(
(
(
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
SNoLt
(
setsum
X6
∅
)
∅
)
)
→
(
(
(
(
(
nat_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
(
(
(
ordinal
X4
∧
(
(
(
exactly4
X5
→
bij
X4
X5
(
λX6 :
set
⇒
∅
)
)
∧
(
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
→
(
¬
SNoEq_
X5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
X5
)
)
)
∧
(
ordinal
X4
→
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
=
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
¬
atleast5
X2
)
)
∧
(
(
(
atleast4
X2
→
reflexive_i
(
λX6 :
set
⇒
λX7 :
set
⇒
(
¬
ordinal
X6
)
)
→
(
¬
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
∈
X4
)
)
)
→
nat_p
∅
)
∧
(
(
ordinal
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
TransSet
X2
)
→
(
(
¬
nat_p
X4
)
∧
atleast4
X4
)
→
(
¬
exactly5
X4
)
)
)
)
)
→
atleast4
X0
→
ordinal
X5
)
∧
(
(
exactly4
X2
→
(
¬
ordinal
X5
)
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
exactly4
X2
)
∧
(
(
¬
exactly4
X2
)
→
set_of_pairs
X4
)
)
)
∧
(
(
¬
atleast3
X4
)
∧
(
(
(
(
exactly5
X4
→
atleastp
X4
∅
)
→
(
¬
setsum_p
X1
)
)
∧
(
¬
(
X5
=
∅
)
)
)
∧
atleast6
X5
)
)
)
→
ordinal
X0
)
→
(
¬
exactly5
X3
)
)
→
(
(
¬
atleast2
X5
)
→
(
¬
exactly2
X0
)
→
(
atleast2
X2
∧
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
)
→
(
¬
binop_on
X5
(
λX6 :
set
⇒
λX7 :
set
⇒
X3
)
)
)
∧
(
atleast6
X0
∧
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∈
X5
)
)
)
→
(
(
X4
∈
X5
)
→
(
¬
exactly3
X3
)
→
exactly2
X5
→
binop_on
(
add_nat
X5
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
(
λX6 :
set
⇒
λX7 :
set
⇒
X6
)
→
(
¬
atleast5
X4
)
)
→
atleast5
X5
→
(
¬
exactly5
X3
)
)
)
∧
(
¬
exactly4
X3
)
)
)
)
→
(
¬
exactly3
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
exactly4
X4
→
exactly5
X3
→
(
¬
TransSet
X3
)
)
→
exactly2
X3
)
∧
(
¬
atleast4
X4
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e3ac72...
and proposition id is
e7c44e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcgdaBGfKewahqjhbs8MKT1m34UvK9z6qw
)
∀X0
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
,
∃X1 ∈
X0
,
(
(
(
∀X2 :
set
,
PNoEq_
X0
(
λX3 :
set
⇒
exactly4
∅
→
(
∃X4 :
set
,
(
(
X4
⊆
X2
)
∧
(
¬
atleast4
X0
)
)
)
)
(
λX3 :
set
⇒
∀X4 :
set
,
(
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
¬
ordinal
X0
)
)
)
)
∧
(
∃X2 ∈
setprod
X1
X1
,
(
¬
trichotomous_or_i
(
λX3 :
set
⇒
λX4 :
set
⇒
(
¬
atleast3
(
⋃
X2
)
)
)
)
)
)
∧
(
∀X2
∈
X1
,
∀X3 :
set
,
∀X4 :
set
,
(
(
(
(
atleast3
X4
∧
(
(
(
¬
nat_p
∅
)
→
nat_p
X3
)
∧
(
¬
TransSet
X2
)
)
)
→
PNoLt
X3
(
λX5 :
set
⇒
(
¬
atleast5
X5
)
)
X0
(
λX5 :
set
⇒
atleast6
X5
)
)
∧
(
(
exactly2
X2
→
(
atleast5
X2
∧
(
(
¬
atleast4
X0
)
∧
exactly4
X3
)
)
)
→
(
ordinal
X3
∧
exactly5
X4
)
)
)
→
(
¬
nat_p
X1
)
→
(
X4
∈
X3
)
)
→
atleast6
X4
)
)
In Proofgold the corresponding term root is
9bd3ac...
and proposition id is
f18690...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMUxAi7EPPgnsqda19SBVyQ3cQAg3Uo2gZ4
)
∃X0 :
set
,
∃X1 :
set
,
(
(
X1
⊆
X0
)
∧
(
(
(
(
∀X2
⊆
X1
,
∀X3 :
set
,
atleast4
X3
→
(
(
∀X4
⊆
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
,
(
¬
exactly2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
(
(
(
∀X4 :
set
,
(
(
¬
SNoLt
X3
X4
)
∧
(
¬
atleast6
∅
)
)
)
∧
(
∃X4 :
set
,
(
(
(
(
¬
nat_p
X0
)
→
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
(
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
∈
X0
)
∧
set_of_pairs
X4
)
∧
(
¬
TransSet
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
¬
TransSet
X4
)
∧
(
¬
atleast6
X4
)
)
)
→
(
(
¬
exactly5
X3
)
→
(
X1
⊆
X3
)
)
→
(
¬
atleast6
X4
)
)
∧
atleast5
X1
)
)
)
→
(
¬
SNo
X1
)
)
)
)
∧
(
∃X2 :
set
,
(
(
X2
⊆
X1
)
∧
(
∃X3 :
set
,
∀X4 :
set
,
(
¬
tuple_p
X3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
(
(
¬
exactly5
X4
)
→
(
(
¬
exactly5
X3
)
∧
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
)
∧
(
(
(
(
exactly3
X4
∧
atleast4
X2
)
∧
(
¬
set_of_pairs
X0
)
)
→
(
(
¬
atleast6
X3
)
∧
(
(
¬
atleast3
(
Sing
∅
)
)
→
(
(
SNo
X3
∧
(
¬
atleast3
∅
)
)
∧
(
(
(
(
exactly3
X4
∧
(
¬
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
¬
exactly4
X3
)
∧
(
¬
SNoLt
X4
X2
)
)
)
∧
(
atleast2
X2
→
(
(
¬
exactly5
X0
)
∧
atleast2
X3
)
)
)
∧
(
¬
atleast3
X0
)
)
)
)
)
→
(
¬
atleast5
X2
)
)
→
(
(
¬
atleast6
X4
)
∧
(
¬
atleast6
X1
)
)
→
atleast4
X4
)
)
)
)
)
)
∧
(
∀X2 :
set
,
∀X3
∈
X1
,
∃X4 :
set
,
(
(
X4
⊆
X1
)
∧
exactly2
X1
)
)
)
→
(
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
∧
PNo_upc
(
λX2 :
set
⇒
λX3 :
set
→
prop
⇒
∃X4 ∈
X1
,
atleast5
X2
)
X1
(
λX2 :
set
⇒
∀X3 :
set
,
(
(
∀X4 :
set
,
(
(
¬
exactly4
X2
)
∧
(
atleast3
X4
→
(
(
eqreln_i
(
λX5 :
set
⇒
λX6 :
set
⇒
atleast4
∅
)
∧
(
(
¬
SNo
X3
)
∧
(
(
¬
TransSet
X4
)
→
(
¬
SNo
X3
)
)
)
)
∧
(
(
¬
ordinal
X0
)
→
TransSet
∅
)
)
)
)
)
∧
(
∀X4 :
set
,
(
(
SNoLe
X2
X3
→
exactly2
X4
)
∧
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly4
X4
)
)
)
→
ordinal
X2
)
)
)
)
In Proofgold the corresponding term root is
174afd...
and proposition id is
43e9b2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMUXeTzGW9gXMcuaxMwUt9HxP8AY9gruzX
)
∀X0
∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
∀X1 :
set
,
∀X2 :
set
,
(
¬
atleast5
X0
)
→
(
∃X3 :
set
,
(
(
¬
setsum_p
X1
)
∧
(
∃X4 :
set
,
(
(
¬
SNo
X2
)
∧
(
(
atleast4
X3
→
(
(
exactly2
X2
→
(
¬
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
∈
∅
)
)
)
→
stricttotalorder_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
(
¬
(
X3
∈
X1
)
)
∧
(
(
(
atleast2
(
V_
∅
)
∧
(
atleast2
X5
→
(
nat_p
∅
∧
SNo
∅
)
)
)
∧
(
¬
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
ordinal
X7
)
)
)
∧
(
(
¬
(
X6
∈
X5
)
)
∧
(
atleast2
X0
∧
atleast6
X0
)
)
)
)
)
)
→
(
¬
setsum_p
X3
)
)
→
(
¬
nat_p
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly4
X2
)
)
)
)
)
)
In Proofgold the corresponding term root is
8ee242...
and proposition id is
96cc88...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMbm9FVPoEuMHStK2svKxfNPrLvo31K3vt4
)
∀X0
⊆
∅
,
∃X1 :
set
,
(
(
X1
⊆
∅
)
∧
(
(
∃X2 ∈
X1
,
exactly5
X2
)
∧
(
∀X2 :
set
,
∀X3
⊆
X2
,
(
(
(
atleast2
X1
∧
(
∃X4 :
set
,
(
(
¬
atleast2
∅
)
∧
ordinal
∅
)
)
)
∧
(
exactly3
X2
→
(
(
X0
∈
X2
)
∧
(
∃X4 ∈
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
,
SNo
X0
)
)
)
)
∧
(
∀X4
⊆
X2
,
(
(
(
(
exactly4
X4
→
(
¬
exactly4
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
∧
(
(
¬
TransSet
X4
)
∧
exactly2
X4
)
)
→
exactly5
X2
)
→
(
atleast5
X4
∧
(
(
¬
atleast4
X4
)
∧
(
atleast6
X0
→
(
¬
TransSet
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
)
→
(
¬
TransSet
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
dc5ee6...
and proposition id is
71bc7e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMMYD3CSnpkbzApiFyit1txz8qx4n3XG3ni
)
∀X0
∈
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
,
∀X1
∈
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∃X2 ∈
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
,
∀X3 :
set
,
(
(
¬
atleast2
X3
)
∧
(
∃X4 :
set
,
(
(
(
¬
nat_p
X1
)
→
(
(
(
¬
TransSet
X2
)
∧
(
exactly2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
→
(
¬
atleast5
X2
)
)
)
∧
(
(
¬
ordinal
X3
)
∧
(
¬
ordinal
X1
)
)
)
)
∧
(
nat_p
∅
→
atleast4
X4
)
)
)
)
→
atleast4
X3
In Proofgold the corresponding term root is
2e9dc4...
and proposition id is
b78a2c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMLfTshnBYQKMWxMNiKKbnmVivnKDX39hDj
)
∃X0 ∈
⋃
∅
,
∀X1
∈
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
,
∀X2
⊆
Inj0
X0
,
∀X3 :
set
,
(
¬
PNoLt
X1
(
λX4 :
set
⇒
(
¬
bij
X2
X1
(
λX5 :
set
⇒
X4
)
)
→
(
(
(
(
(
(
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
∧
(
(
¬
atleast4
X3
)
∧
(
(
¬
atleast6
X2
)
∧
(
(
TransSet
∅
→
(
(
(
(
bij
X2
X3
(
λX5 :
set
⇒
X4
)
→
(
¬
exactly5
X1
)
→
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
→
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
→
ordinal
X3
)
→
(
¬
TransSet
X3
)
)
→
(
atleast2
∅
∧
(
set_of_pairs
X3
∧
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
)
→
atleast3
X3
)
)
)
)
→
exactly2
X2
)
∧
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
∧
(
(
(
atleast2
X3
∧
(
(
(
¬
atleast3
X0
)
→
(
(
(
(
(
(
(
¬
atleast4
X3
)
→
(
ordinal
∅
∧
(
¬
atleast5
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
)
∧
(
(
¬
(
Sing
X2
∈
X1
)
)
∧
(
(
¬
exactly2
X3
)
∧
(
(
¬
exactly5
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
(
¬
(
X3
=
X4
)
)
)
)
)
)
→
atleast3
X4
→
atleast6
X1
)
→
(
¬
atleast6
X3
)
)
→
nat_p
X3
)
∧
(
(
¬
atleast6
∅
)
→
(
¬
exactly2
X1
)
)
)
)
∧
(
(
¬
atleast3
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
symmetric_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
atleast5
X5
)
→
(
(
(
atleast3
X6
→
(
(
¬
reflexive_i
(
λX7 :
set
⇒
λX8 :
set
⇒
atleast4
X7
)
)
→
(
¬
exactly4
X6
)
)
→
(
¬
exactly5
X0
)
→
TransSet
(
setminus
X2
X5
)
)
→
(
(
(
¬
exactly5
X5
)
→
(
(
¬
atleastp
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
X6
)
∧
(
(
(
(
(
¬
nat_p
X4
)
→
(
¬
SNoLt
X6
X6
)
)
→
ordinal
∅
)
→
(
¬
atleast3
X6
)
)
∧
(
¬
ordinal
X5
)
)
)
)
∧
(
(
(
(
(
exactly4
(
UPair
X6
X5
)
→
(
TransSet
X3
∧
(
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
→
(
(
(
(
(
¬
(
X5
=
SetAdjoin
X3
X4
)
)
→
exactly5
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
→
(
(
(
atleast6
X0
→
(
exactly3
X6
∧
(
nat_p
X6
∧
(
(
(
(
(
(
(
(
¬
atleast4
∅
)
→
atleast5
X5
)
∧
(
antisymmetric_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
(
exactly5
X7
→
(
¬
exactly3
X8
)
→
(
¬
nat_p
X7
)
)
∧
(
¬
(
X2
∈
X7
)
)
)
)
→
(
¬
atleast5
X6
)
)
)
∧
(
¬
atleast6
X5
)
)
∧
(
(
(
(
(
¬
exactly4
X6
)
∧
(
set_of_pairs
X5
∧
(
¬
TransSet
X6
)
)
)
∧
(
(
SNo
X2
→
(
(
(
¬
tuple_p
X6
X5
)
∧
(
(
¬
exactly4
X1
)
→
(
(
atleast3
X5
∧
(
atleast5
X6
∧
(
(
exactly5
X0
→
(
(
¬
nat_p
X5
)
∧
(
¬
SNoLt
X6
X0
)
)
)
→
atleast4
X0
)
)
)
∧
TransSet
X0
)
)
)
∧
exactly3
∅
)
)
→
(
(
(
¬
atleast2
(
Inj1
X3
)
)
→
ordinal
∅
→
(
TransSet
X6
∧
(
exactly4
X6
→
(
¬
SNoLt
X1
∅
)
)
)
)
→
(
(
(
(
¬
atleast5
X6
)
→
exactly4
X5
)
∧
(
(
¬
nat_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
setsum_p
X5
)
)
∧
(
exactly2
X0
∧
(
atleast4
X0
∧
(
(
¬
exactly3
X5
)
∧
(
(
(
(
(
¬
atleast6
X1
)
∧
atleast6
X6
)
→
(
¬
ordinal
(
ordsucc
X6
)
)
)
∧
(
(
exactly2
X6
∧
(
¬
nat_p
X1
)
)
∧
atleast5
X5
)
)
→
(
(
(
¬
atleast6
X0
)
→
(
atleast5
X5
∧
(
(
¬
ordinal
∅
)
→
exactly5
X5
→
(
(
(
(
(
¬
atleast3
X2
)
→
(
TransSet
X6
∧
(
¬
ordinal
X4
)
)
)
→
(
(
(
¬
exactly3
X6
)
→
(
¬
exactly2
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
)
∧
(
atleast6
X0
∧
SNoEq_
X6
X5
X3
)
)
)
∧
(
(
¬
atleast2
(
SNoLev
X5
)
)
→
(
¬
tuple_p
X6
X6
)
)
)
∧
(
(
(
(
¬
SNoLt
X6
X0
)
→
(
(
(
(
(
(
¬
ordinal
X6
)
→
(
(
(
exactly4
X5
∧
(
¬
exactly3
X5
)
)
∧
(
(
setsum_p
X5
∧
(
(
(
(
¬
atleast4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
(
(
(
(
exactly5
X6
∧
(
(
¬
set_of_pairs
X6
)
∧
(
(
¬
exactly3
X3
)
∧
(
¬
TransSet
X5
)
)
)
)
→
bij
X5
X6
(
λX7 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
∧
(
¬
atleast4
X5
)
)
→
atleast4
X0
)
∧
(
¬
exactly4
X3
)
)
)
∧
(
(
¬
tuple_p
X2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
(
¬
setsum_p
X5
)
)
)
∧
(
nat_p
X5
→
(
¬
atleast5
X1
)
)
)
)
∧
TransSet
X0
)
)
∧
PNoLt
X5
(
λX7 :
set
⇒
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
→
(
(
(
¬
TransSet
∅
)
→
(
(
(
(
(
¬
TransSet
X7
)
∧
(
(
(
¬
TransSet
X1
)
→
(
(
¬
atleast6
X7
)
∧
(
atleast3
X7
→
(
atleast2
X7
→
(
¬
atleast2
X2
)
→
SNo
X7
)
→
(
exactly2
X7
→
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
→
(
(
¬
exactly4
X7
)
∧
(
¬
SNo
X0
)
)
)
)
)
∧
(
(
(
nat_p
∅
→
atleast5
(
ordsucc
X6
)
)
→
SNo_
X3
X7
)
→
(
¬
exactly2
X1
)
)
)
)
∧
atleast4
(
Inj0
(
Inj1
X7
)
)
)
→
(
(
(
SNo
X1
∧
atleast5
∅
)
∧
atleast6
X6
)
∧
(
(
X0
⊆
X0
)
→
(
(
¬
exactly3
X4
)
∧
(
¬
exactly2
X6
)
)
)
)
)
→
(
(
(
(
(
atleast4
X6
∧
(
(
¬
exactly5
X6
)
∧
partialorder_i
(
λX8 :
set
⇒
λX9 :
set
⇒
exactly3
X3
)
)
)
∧
(
(
¬
SNoEq_
X2
X6
∅
)
→
(
¬
atleast3
X7
)
)
)
∧
exactly4
X6
)
→
(
TransSet
X7
∧
(
(
¬
linear_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
(
¬
irreflexive_i
(
λX10 :
set
⇒
λX11 :
set
⇒
setsum_p
X11
)
)
∧
(
(
(
(
(
X8
∈
⋃
X9
)
∧
(
(
¬
exactly4
X0
)
∧
(
(
¬
exactly3
X3
)
→
(
atleast6
X2
∧
(
(
(
(
¬
atleast5
X5
)
→
nat_p
X1
)
∧
(
¬
nat_p
X0
)
)
∧
(
¬
exactly5
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
)
→
(
¬
strictpartialorder_i
(
λX10 :
set
⇒
λX11 :
set
⇒
(
(
atleast2
∅
→
(
¬
atleast2
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
∧
exactly5
∅
)
)
)
)
)
)
∧
atleast4
X9
)
→
(
(
(
(
¬
atleast3
X8
)
∧
(
¬
atleast6
X9
)
)
→
atleast5
X8
)
∧
(
¬
TransSet
X8
)
)
)
∧
(
(
(
¬
nat_p
X6
)
→
(
¬
TransSet
X0
)
)
→
ordinal
X8
→
(
(
atleast2
X6
→
nat_p
∅
→
TransSet
X9
)
∧
(
¬
ordinal
X9
)
)
)
)
)
)
)
∧
(
¬
atleast2
X6
)
)
)
→
(
¬
tuple_p
(
Inj0
X6
)
X6
)
)
∧
atleast3
X2
)
→
ordinal
X5
)
→
(
¬
ordinal
X0
)
)
→
(
(
X1
⊆
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
∧
(
(
¬
atleast4
X3
)
∧
(
¬
SNoLt
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
X6
)
)
)
)
→
(
(
(
(
¬
nat_p
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
(
(
¬
setsum_p
X2
)
∧
(
¬
atleast6
X7
)
)
)
→
(
(
exactly3
X7
→
(
(
¬
reflexive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
(
(
(
(
(
(
¬
exactly3
X8
)
∧
(
(
(
X8
∈
X4
)
∧
TransSet
X4
)
→
(
(
(
SNo
X1
→
(
(
¬
SNo
X4
)
→
(
¬
atleast6
X9
)
)
→
atleast6
X9
)
→
(
¬
setsum_p
X2
)
)
∧
(
(
atleast4
∅
→
(
¬
atleast3
(
𝒫
X9
)
)
)
→
atleast2
X9
)
)
)
)
→
(
¬
atleastp
X9
X8
)
)
→
exactly4
X9
)
→
exactly3
X9
)
→
(
(
(
(
¬
nat_p
X9
)
→
(
¬
exactly4
X0
)
)
→
(
(
(
¬
setsum_p
X9
)
∧
exactly2
X9
)
∧
(
(
(
(
¬
setsum_p
X9
)
→
exactly2
∅
)
→
(
(
¬
atleast2
X8
)
∧
(
¬
exactly3
X4
)
)
)
→
(
¬
exactly5
X1
)
)
)
)
∧
(
(
(
(
ordinal
X8
∧
(
¬
setsum_p
X8
)
)
∧
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
∧
(
¬
atleast6
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
)
∧
exactly2
X1
)
)
)
∧
atleast6
X9
)
)
)
∧
(
TransSet
X7
→
(
¬
setsum_p
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
)
→
atleast6
X1
)
→
(
¬
transitive_i
(
λX8 :
set
⇒
λX9 :
set
⇒
(
¬
atleast6
X4
)
→
(
atleast6
X9
∧
setsum_p
X0
)
→
reflexive_i
(
λX10 :
set
⇒
λX11 :
set
⇒
atleast2
X9
)
→
(
(
¬
atleast5
X6
)
∧
(
¬
atleast2
X8
)
)
)
)
)
∧
(
(
(
(
TransSet
X7
∧
atleast3
X0
)
→
(
¬
atleast6
X6
)
)
→
(
¬
TransSet
X7
)
)
→
exactly4
X7
)
)
)
X5
(
λX7 :
set
⇒
(
exactly3
X7
∧
(
atleast2
X6
∧
(
¬
setsum_p
X0
)
)
)
)
)
)
∧
(
(
(
¬
atleast6
X5
)
∧
(
¬
exactly3
(
add_nat
∅
X0
)
)
)
→
exactly5
X0
→
(
¬
exactly2
X6
)
)
)
→
atleast6
X5
)
→
(
X6
∈
X6
)
)
∧
(
(
¬
exactly5
X6
)
∧
(
¬
SNo
X6
)
)
)
→
(
¬
ordinal
X5
)
→
(
¬
atleast2
X5
)
→
(
(
(
¬
SNo
X0
)
∧
(
(
(
¬
(
X0
∈
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
)
→
atleast3
X6
)
∧
(
(
¬
atleast2
X4
)
→
binop_on
X6
(
λX7 :
set
⇒
λX8 :
set
⇒
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
)
)
)
∧
(
exactly3
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
SNo
X5
→
setsum_p
X5
→
(
¬
set_of_pairs
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
trichotomous_or_i
(
λX7 :
set
⇒
λX8 :
set
⇒
(
¬
exactly4
X6
)
)
)
)
)
∧
(
¬
atleast2
∅
)
)
→
(
¬
atleast6
X0
)
)
)
)
)
→
(
(
set_of_pairs
X5
∧
(
(
¬
TransSet
X5
)
→
atleast6
X0
)
)
∧
(
(
(
(
(
(
¬
PNoEq_
X3
(
λX7 :
set
⇒
(
¬
exactly4
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
∅
)
)
)
(
λX7 :
set
⇒
SNo_
∅
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
∅
)
)
)
→
SNoLt
X6
X4
)
∧
(
¬
atleast2
X6
)
)
→
exactly3
(
Inj0
X5
)
)
→
(
(
atleast4
(
proj1
X4
)
∧
(
¬
(
X5
∈
X5
)
)
)
∧
(
¬
atleast4
X5
)
)
)
→
exactly2
(
setminus
X5
X3
)
)
)
)
∧
(
¬
(
X6
∈
X1
)
)
)
)
)
)
)
)
)
→
(
¬
atleast5
X5
)
)
)
→
(
¬
atleast3
X5
)
)
∧
exactly2
X0
)
)
∧
(
X3
⊆
X5
)
)
→
SNo
X1
)
∧
atleast3
X6
)
)
)
)
→
tuple_p
X6
X6
)
∧
(
equip
X0
X6
∧
(
(
¬
atleast2
X0
)
∧
(
(
(
¬
nat_p
X5
)
→
TransSet
X6
)
→
(
¬
(
X5
∈
X2
)
)
)
)
)
)
)
→
(
(
¬
atleast3
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
)
∧
(
¬
atleast5
∅
)
)
)
∧
(
(
(
(
(
¬
tuple_p
X5
X5
)
∧
(
¬
ordinal
X6
)
)
∧
exactly5
X5
)
→
set_of_pairs
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
)
→
(
atleast3
X5
∧
(
atleast6
X6
→
(
¬
atleast5
X6
)
)
)
→
setsum_p
X5
→
(
¬
exactly2
X5
)
→
exactly2
(
binrep
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast6
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
→
atleast4
X6
)
)
∧
(
¬
atleast6
X6
)
)
)
)
)
∧
(
¬
exactly2
X6
)
)
∧
(
¬
atleast5
(
binunion
X1
X2
)
)
)
∧
atleast2
X0
)
→
(
(
(
(
(
(
¬
ordinal
X5
)
→
(
¬
atleast4
X5
)
)
∧
(
TransSet
X6
→
(
¬
setsum_p
X5
)
)
)
∧
(
(
(
(
¬
(
X6
∈
X1
)
)
→
exactly5
(
setprod
X4
X3
)
→
exactly5
X5
→
PNoEq_
(
Inj0
(
binintersect
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
)
)
(
λX7 :
set
⇒
nat_p
(
⋃
X6
)
)
(
λX7 :
set
⇒
(
¬
exactly5
X6
)
)
)
→
(
¬
atleast6
X6
)
)
∧
(
¬
exactly5
X5
)
)
)
→
(
¬
atleast3
X0
)
)
∧
(
(
¬
atleastp
X5
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
∧
(
¬
atleast3
X4
)
)
)
)
)
)
∧
exactly5
X2
)
)
→
(
∅
∈
X2
)
)
)
)
→
nat_p
X4
)
∧
(
¬
atleast2
X3
)
)
)
∧
(
(
¬
exactly2
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
¬
exactly4
X3
)
)
)
→
(
exactly2
X3
→
(
¬
nat_p
X3
)
)
→
exactly3
X4
)
→
(
bij
X3
X4
(
λX5 :
set
⇒
X5
)
∧
(
(
(
(
(
(
(
nat_p
X2
→
(
¬
exactly2
X3
)
)
→
(
¬
atleast3
X4
)
→
(
¬
ordinal
X4
)
)
∧
(
(
exactly4
X3
→
exactly4
X4
)
→
setsum_p
X1
→
(
(
(
atleast3
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
∅
)
→
(
¬
atleast4
X4
)
)
∧
(
(
(
(
(
¬
exactly3
X2
)
∧
(
¬
atleast2
(
binrep
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
∅
)
)
)
→
(
atleast4
∅
∧
(
¬
atleast4
X0
)
)
)
→
reflexive_i
(
λX5 :
set
⇒
λX6 :
set
⇒
(
¬
ordinal
(
binunion
X5
X0
)
)
)
)
→
(
¬
exactly3
X3
)
)
)
∧
(
atleast3
(
UPair
X4
X2
)
∧
(
exactly5
X3
∧
(
atleast5
X1
→
(
¬
TransSet
X3
)
)
)
)
)
)
)
∧
(
¬
SNo
X4
)
)
→
(
⋃
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
∅
)
)
=
X4
)
)
∧
(
¬
exactly2
(
binrep
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
(
𝒫
(
𝒫
∅
)
)
)
)
)
→
(
X4
∈
X2
)
→
(
¬
atleast2
X2
)
)
)
)
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
∅
)
)
(
λX4 :
set
⇒
atleast3
X2
)
)
In Proofgold the corresponding term root is
ab8e62...
and proposition id is
93cb60...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_Random1_TMcECSsBjuKKHaiVUngwnNXxHPbv584cXv5
)
∃X0 :
set
,
∀X1 :
set
,
(
(
∀X2 :
set
,
exactly5
X0
→
(
∃X3 :
set
,
(
(
X3
⊆
∅
)
∧
(
atleast5
X1
→
transitive_i
(
λX4 :
set
⇒
λX5 :
set
⇒
(
(
¬
TransSet
(
binrep
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
∅
)
)
→
atleast4
X5
)
→
(
(
SNo_
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
∅
)
X2
→
(
(
(
¬
exactly2
(
V_
X4
)
)
∧
atleast2
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
∧
atleast3
X1
)
)
∧
atleast4
X0
)
)
)
)
)
)
→
(
¬
SNo_
X1
X0
)
)
→
(
¬
exactly1of2
(
(
(
∀X2 :
set
,
∃X3 ∈
ordsucc
X0
,
(
∀X4 :
set
,
(
(
¬
atleast3
(
𝒫
(
𝒫
(
𝒫
(
𝒫
∅
)
)
)
)
)
∧
(
SNoEq_
X4
X4
X0
∧
(
¬
setsum_p
(
binrep
(
binrep
(
𝒫
(
binrep
(
𝒫
(
𝒫
∅
)
)
∅
)
)
(
𝒫
(
𝒫
∅
)
)
)
(
𝒫
∅
)
)
)
)
)
)
→
(
¬
atleast5
X3
)
)
∧
(
∃X2 :
set
,
∃X3 :
set
,
(
¬
atleast3
X3
)
)
)
∧
(
∀X2
⊆
X0
,
TransSet
X2
→
(
∀X3
∈
X0
,
∃X4 :
set
,
(
(
X4
⊆
X3
)
∧
(
¬
ordinal
X4
)
)
)
)
)
(
∀X2 :
set
,
(
¬
exactly2
X2
)
→
(
∃X3 :
set
,
∃X4 :
set
,
(
(
SNo
∅
→
atleast6
X4
)
∧
exactly3
X3
)
)
)
)
In Proofgold the corresponding term root is
958baa...
and proposition id is
37b4d0...
Proof:
The rest of this subproof is missing.
∎
End of Section
Random1