Primitive. The name
Eps_i is a term of type
(set → prop) → set.
L2Axiom. (
Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
L3Definition. We define
True to be
∀p : prop, p → p of type
prop.
L4Definition. We define
False to be
∀p : prop, p of type
prop.
L5Definition. We define
not to be
λA : prop ⇒ A → False of type
prop → prop.
Notation. We use
¬ as a prefix operator with priority 700 corresponding to applying term
not.
L8Definition. We define
and to be
λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type
prop → prop → prop.
Notation. We use
∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term
and.
L11Definition. We define
or to be
λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type
prop → prop → prop.
Notation. We use
∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term
or.
L14Definition. We define
iff to be
λA B : prop ⇒ and (A → B) (B → A) of type
prop → prop → prop.
Notation. We use
↔ as an infix operator with priority 805 and no associativity corresponding to applying term
iff.
Beginning of Section Eq
L19Definition. We define
eq to be
λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type
A → A → prop.
L20Definition. We define
neq to be
λx y : A ⇒ ¬ eq x y of type
A → A → prop.
End of Section Eq
Notation. We use
= as an infix operator with priority 502 and no associativity corresponding to applying term
eq.
Notation. We use
≠ as an infix operator with priority 502 and no associativity corresponding to applying term
neq.
Beginning of Section FE
L27Axiom. (
func_ext) We take the following as an axiom:
∀f g : A → B, (∀x : A, f x = g x) → f = g
End of Section FE
Beginning of Section Ex
L31Definition. We define
ex to be
λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type
(A → prop) → prop.
End of Section Ex
Notation. We use
∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex.
L35Axiom. (
prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
Primitive. The name
In is a term of type
set → set → prop.
Notation. We use
∈ as an infix operator with priority 500 and no associativity corresponding to applying term
In. Furthermore, we may write
∀ x ∈ A, B to mean
∀ x : set, x ∈ A → B.
L37Definition. We define
Subq to be
λA B ⇒ ∀x ∈ A, x ∈ B of type
set → set → prop.
Notation. We use
⊆ as an infix operator with priority 500 and no associativity corresponding to applying term
Subq. Furthermore, we may write
∀ x ⊆ A, B to mean
∀ x : set, x ⊆ A → B.
L38Axiom. (
set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
L39Axiom. (
In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
Notation. We use
∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex and handling ∈ or ⊆ ascriptions using
and.
Primitive. The name
Empty is a term of type
set.
L42Axiom. (
EmptyAx) We take the following as an axiom:
Primitive. The name
⋃ is a term of type
set → set.
L45Axiom. (
UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
Primitive. The name
𝒫 is a term of type
set → set.
L48Axiom. (
PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
Primitive. The name
Repl is a term of type
set → (set → set) → set.
Notation.
{B| x ∈ A} is notation for
Repl A (λ x . B).
L51Axiom. (
ReplEq) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x
L52Definition. We define
TransSet to be
λU : set ⇒ ∀x ∈ U, x ⊆ U of type
set → prop.
L53Definition. We define
Union_closed to be
λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type
set → prop.
L54Definition. We define
Power_closed to be
λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type
set → prop.
L55Definition. We define
Repl_closed to be
λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type
set → prop.
Primitive. The name
UnivOf is a term of type
set → set.
L62Axiom. (
UnivOf_In) We take the following as an axiom:
L65Axiom. (
UnivOf_Min) We take the following as an axiom:
L69Axiom. (
andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
L71Axiom. (
orIL) We take the following as an axiom:
L73Axiom. (
orIR) We take the following as an axiom:
L75Axiom. (
iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
L77Axiom. (
pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
L79Definition. We define
nIn to be
λx X ⇒ ¬ In x X of type
set → set → prop.
Notation. We use
∉ as an infix operator with priority 502 and no associativity corresponding to applying term
nIn.
L84Axiom. (
EmptyE) We take the following as an axiom:
L86Axiom. (
PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
L88Axiom. (
Subq_Empty) We take the following as an axiom:
L92Axiom. (
xm) We take the following as an axiom:
L94Axiom. (
FalseE) We take the following as an axiom:
L96Axiom. (
andEL) We take the following as an axiom:
L98Axiom. (
andER) We take the following as an axiom:
Beginning of Section PropN
L102Variable P1 P2 P3 : prop
L103Axiom. (
and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
L105Axiom. (
and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
L107Axiom. (
or3I1) We take the following as an axiom:
L109Axiom. (
or3I2) We take the following as an axiom:
L111Axiom. (
or3I3) We take the following as an axiom:
L113Axiom. (
or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
L117Axiom. (
and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
L121Axiom. (
and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
L125Axiom. (
and6I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P6 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6
L129Axiom. (
and7I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P6 → P7 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7
End of Section PropN
L137Axiom. (
iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
L139Axiom. (
iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
L141Axiom. (
iff_refl) We take the following as an axiom:
L143Axiom. (
iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
L145Axiom. (
iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
L147Axiom. (
eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
L149Axiom. (
neq_i_sym) We take the following as an axiom:
L151Axiom. (
Eps_i_ex) We take the following as an axiom:
∀P : set → prop, (∃x, P x) → P (Eps_i P)
L153Axiom. (
prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
L155Axiom. (
Subq_ref) We take the following as an axiom:
L157Axiom. (
Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
L161Axiom. (
Empty_eq) We take the following as an axiom:
L163Axiom. (
UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
L165Axiom. (
UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
L167Axiom. (
UnionE_impred) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
L169Axiom. (
PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
L173Axiom. (
dneg) We take the following as an axiom:
L177Axiom. (
eq_or_nand) We take the following as an axiom:
L181Definition. We define
exactly1of2 to be
λA B : prop ⇒ A ∧ ¬ B ∨ ¬ A ∧ B of type
prop → prop → prop.
L187Axiom. (
exactly1of2_E) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
L195Axiom. (
ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
L197Axiom. (
ReplE) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x
L199Axiom. (
ReplE_impred) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p
L201Axiom. (
ReplE') We take the following as an axiom:
∀X, ∀f : set → set, ∀p : set → prop, (∀x ∈ X, p (f x)) → ∀y ∈ {f x|x ∈ X}, p y
L203Axiom. (
Repl_Empty) We take the following as an axiom:
L205Axiom. (
ReplEq_ext_sub) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}
L207Axiom. (
ReplEq_ext) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
L209Axiom. (
Repl_inv_eq) We take the following as an axiom:
∀P : set → prop, ∀f g : set → set, (∀x, P x → g (f x) = x) → ∀X, (∀x ∈ X, P x) → {g y|y ∈ {f x|x ∈ X}} = X
L213Axiom. (
Repl_invol_eq) We take the following as an axiom:
∀P : set → prop, ∀f : set → set, (∀x, P x → f (f x) = x) → ∀X, (∀x ∈ X, P x) → {f y|y ∈ {f x|x ∈ X}} = X
L219Definition. We define
If_i to be
(λp x y ⇒ Eps_i (λz : set ⇒ p ∧ z = x ∨ ¬ p ∧ z = y)) of type
prop → set → set → set.
Notation.
if cond then T else E is notation corresponding to
If_i type cond T E where
type is the inferred type of
T.
L224Axiom. (
If_i_0) We take the following as an axiom:
L227Axiom. (
If_i_1) We take the following as an axiom:
L230Axiom. (
If_i_or) We take the following as an axiom:
Notation.
{x,y} is notation for
UPair x y.
L237Axiom. (
UPairE) We take the following as an axiom:
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
L240Axiom. (
UPairI1) We take the following as an axiom:
L242Axiom. (
UPairI2) We take the following as an axiom:
L246Definition. We define
Sing to be
λx ⇒ {x,x} of type
set → set.
Notation.
{x} is notation for
Sing x.
L248Axiom. (
SingI) We take the following as an axiom:
L250Axiom. (
SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
L252Definition. We define
binunion to be
λX Y ⇒ ⋃ {X,Y} of type
set → set → set.
Notation. We use
∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term
binunion.
L255Axiom. (
binunionI1) We take the following as an axiom:
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
L257Axiom. (
binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
L259Axiom. (
binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
L261Axiom. (
binunionE') We take the following as an axiom:
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
L263Axiom. (
binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
L267Axiom. (
binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
L281Definition. We define
SetAdjoin to be
λX y ⇒ X ∪ {y} of type
set → set → set.
Notation. We now use the set enumeration notation
{...,...,...} in general. If 0 elements are given, then
Empty is used to form the corresponding term. If 1 element is given, then
Sing is used to form the corresponding term. If 2 elements are given, then
UPair is used to form the corresponding term. If more than elements are given, then
SetAdjoin is used to reduce to the case with one fewer elements.
L285Definition. We define
famunion to be
λX F ⇒ ⋃ {F x|x ∈ X} of type
set → (set → set) → set.
Notation. We use
⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
famunion.
L290Axiom. (
famunionI) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃x ∈ XF x
L292Axiom. (
famunionE) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
L294Axiom. (
famunionE_impred) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p
Beginning of Section SepSec
L305Variable P : set → prop
L306Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
End of Section SepSec
Notation.
{x ∈ A | B} is notation for
Sep A (λ x . B).
L313Axiom. (
SepI) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
L315Axiom. (
SepE) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
L317Axiom. (
SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
L319Axiom. (
SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
L321Axiom. (
Sep_Empty) We take the following as an axiom:
L323Axiom. (
Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
L325Axiom. (
Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
L329Definition. We define
ReplSep to be
λX P F ⇒ {F x|x ∈ {z ∈ X|P z}} of type
set → (set → prop) → (set → set) → set.
Notation.
{B| x ∈ A, C} is notation for
ReplSep A (λ x . C) (λ x . B).
L331Axiom. (
ReplSepI) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}
L333Axiom. (
ReplSepE) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x
L335Axiom. (
ReplSepE_impred) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p
L339Definition. We define
binintersect to be
λX Y ⇒ {x ∈ X|x ∈ Y} of type
set → set → set.
Notation. We use
∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term
binintersect.
L343Axiom. (
binintersectI) We take the following as an axiom:
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
L345Axiom. (
binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
L365Definition. We define
setminus to be
λX Y ⇒ Sep X (λx ⇒ x ∉ Y) of type
set → set → set.
Notation. We use
∖ as an infix operator with priority 350 and no associativity corresponding to applying term
setminus.
L369Axiom. (
setminusI) We take the following as an axiom:
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
L371Axiom. (
setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
L373Axiom. (
setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
L375Axiom. (
setminusE2) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∉ Y
L383Axiom. (
In_irref) We take the following as an axiom:
L385Axiom. (
In_no2cycle) We take the following as an axiom:
L389Definition. We define
ordsucc to be
λx : set ⇒ x ∪ {x} of type
set → set.
L390Axiom. (
ordsuccI1) We take the following as an axiom:
L392Axiom. (
ordsuccI2) We take the following as an axiom:
L394Axiom. (
ordsuccE) We take the following as an axiom:
Notation. Natural numbers 0,1,2,... are notation for the terms formed using
Empty as 0 and forming successors with
ordsucc.
L402Axiom. (
ordsucc_inj) We take the following as an axiom:
L404Axiom. (
In_0_1) We take the following as an axiom:
L406Axiom. (
In_0_2) We take the following as an axiom:
L408Axiom. (
In_1_2) We take the following as an axiom:
L410Definition. We define
nat_p to be
λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type
set → prop.
L412Axiom. (
nat_0) We take the following as an axiom:
L414Axiom. (
nat_ordsucc) We take the following as an axiom:
L416Axiom. (
nat_1) We take the following as an axiom:
L418Axiom. (
nat_2) We take the following as an axiom:
L424Axiom. (
nat_ind) We take the following as an axiom:
L430Axiom. (
nat_inv) We take the following as an axiom:
L432Axiom. (
nat_p_trans) We take the following as an axiom:
L434Axiom. (
nat_trans) We take the following as an axiom:
L438Definition. We define
surj to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type
set → set → (set → set) → prop.
L446Definition. We define
inj to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type
set → set → (set → set) → prop.
L454Axiom. (
injI) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀x ∈ X, f x ∈ Y) → (∀x z ∈ X, f x = f z → x = z) → inj X Y f
L456Axiom. (
inj_comp) We take the following as an axiom:
∀X Y Z : set, ∀f g : set → set, inj X Y f → inj Y Z g → inj X Z (λx ⇒ g (f x))
L458Definition. We define
bij to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type
set → set → (set → set) → prop.
L466Axiom. (
bijI) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f
L472Axiom. (
bijE) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p
L481Axiom. (
bij_inj) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → inj X Y f
L483Axiom. (
bij_id) We take the following as an axiom:
L485Axiom. (
bij_comp) We take the following as an axiom:
∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))
L487Axiom. (
bij_surj) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → surj X Y f
L489Definition. We define
inv to be
λX f ⇒ λy : set ⇒ Eps_i (λx ⇒ x ∈ X ∧ f x = y) of type
set → (set → set) → set → set.
L491Axiom. (
surj_rinv) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y
L493Axiom. (
inj_linv) We take the following as an axiom:
∀X, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x
L495Axiom. (
bij_inv) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
L497Definition. We define
atleastp to be
λX Y : set ⇒ ∃f : set → set, inj X Y f of type
set → set → prop.
L504Definition. We define
equip to be
λX Y : set ⇒ ∃f : set → set, bij X Y f of type
set → set → prop.
L509Axiom. (
equip_ref) We take the following as an axiom:
L511Axiom. (
equip_sym) We take the following as an axiom:
L513Axiom. (
equip_tra) We take the following as an axiom:
Beginning of Section SchroederBernstein
L523Axiom. (
KnasterTarski_set) We take the following as an axiom:
∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y
L528Axiom. (
image_In_Power) We take the following as an axiom:
∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B
End of Section SchroederBernstein
Beginning of Section PigeonHole
End of Section PigeonHole
L550Axiom. (
cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
L552Axiom. (
cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
L554Axiom. (
neq_0_1) We take the following as an axiom:
L556Axiom. (
neq_1_0) We take the following as an axiom:
L558Axiom. (
neq_0_2) We take the following as an axiom:
L560Axiom. (
neq_2_0) We take the following as an axiom:
L576Axiom. (
ordinal_1) We take the following as an axiom:
L578Axiom. (
ordinal_2) We take the following as an axiom:
L605Axiom. (
ordinal_ind) We take the following as an axiom:
∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha
L616Axiom. (
exandE_i) We take the following as an axiom:
∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r
L618Axiom. (
exandE_ii) We take the following as an axiom:
∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p
L620Axiom. (
exandE_iii) We take the following as an axiom:
∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p
L622Axiom. (
exandE_iiii) We take the following as an axiom:
∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p
Beginning of Section Descr_ii
L626Variable P : (set → set) → prop
L628Definition. We define
Descr_ii to be
λx : set ⇒ Eps_i (λy ⇒ ∀h : set → set, P h → h x = y) of type
set → set.
L629Hypothesis Pex : ∃f : set → set, P f
L630Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
End of Section Descr_ii
Beginning of Section Descr_iii
L637Variable P : (set → set → set) → prop
L639Definition. We define
Descr_iii to be
λx y : set ⇒ Eps_i (λz ⇒ ∀h : set → set → set, P h → h x y = z) of type
set → set → set.
L640Hypothesis Pex : ∃f : set → set → set, P f
L641Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
End of Section Descr_iii
Beginning of Section Descr_Vo1
L650Definition. We define
Descr_Vo1 to be
λx : set ⇒ ∀h : set → prop, P h → h x of type
Vo 1.
L651Hypothesis Pex : ∃f : Vo 1, P f
L652Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
End of Section Descr_Vo1
Beginning of Section If_ii
L660Variable f g : set → set
L662Definition. We define
If_ii to be
λx ⇒ if p then f x else g x of type
set → set.
L663Axiom. (
If_ii_1) We take the following as an axiom:
L665Axiom. (
If_ii_0) We take the following as an axiom:
End of Section If_ii
Beginning of Section If_iii
L672Variable f g : set → set → set
L674Definition. We define
If_iii to be
λx y ⇒ if p then f x y else g x y of type
set → set → set.
L675Axiom. (
If_iii_1) We take the following as an axiom:
L677Axiom. (
If_iii_0) We take the following as an axiom:
End of Section If_iii
Beginning of Section EpsilonRec_i
L683Variable F : set → (set → set) → set
L684Definition. We define
In_rec_i_G to be
λX Y ⇒ ∀R : set → set → prop, (∀X : set, ∀f : set → set, (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type
set → set → prop.
L695Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
L703Axiom. (
In_rec_i_eq) We take the following as an axiom:
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
L709Variable F : set → (set → (set → set)) → (set → set)
L710Definition. We define
In_rec_G_ii to be
λX Y ⇒ ∀R : set → (set → set) → prop, (∀X : set, ∀f : set → (set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type
set → (set → set) → prop.
L721Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
L735Variable F : set → (set → (set → set → set)) → (set → set → set)
L736Definition. We define
In_rec_G_iii to be
λX Y ⇒ ∀R : set → (set → set → set) → prop, (∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type
set → (set → set → set) → prop.
L747Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_iii
Beginning of Section NatRec
L762Variable f : set → set → set
L763Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
L765Axiom. (
nat_primrec_r) We take the following as an axiom:
∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section NatRec
Beginning of Section NatAdd
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_nat.
L778Axiom. (
add_nat_0R) We take the following as an axiom:
L780Axiom. (
add_nat_SR) We take the following as an axiom:
L782Axiom. (
add_nat_p) We take the following as an axiom:
L788Axiom. (
add_nat_0L) We take the following as an axiom:
L790Axiom. (
add_nat_SL) We take the following as an axiom:
L792Axiom. (
add_nat_com) We take the following as an axiom:
End of Section NatAdd
Beginning of Section NatMul
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_nat.
L814Definition. We define
mul_nat to be
λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type
set → set → set.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_nat.
L817Axiom. (
mul_nat_0R) We take the following as an axiom:
L819Axiom. (
mul_nat_SR) We take the following as an axiom:
L821Axiom. (
mul_nat_1R) We take the following as an axiom:
L823Axiom. (
mul_nat_p) We take the following as an axiom:
L825Axiom. (
mul_nat_0L) We take the following as an axiom:
L827Axiom. (
mul_nat_SL) We take the following as an axiom:
L829Axiom. (
mul_nat_com) We take the following as an axiom:
L848Definition. We define
Pi_nat to be
λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type
(set → set) → set → set.
L851Axiom. (
Pi_nat_0) We take the following as an axiom:
L853Axiom. (
Pi_nat_S) We take the following as an axiom:
L855Axiom. (
Pi_nat_p) We take the following as an axiom:
End of Section NatMul
L868Axiom. (
ZF_closed_E) We take the following as an axiom:
L902Axiom. (
omega_nat_p) We take the following as an axiom:
L904Axiom. (
nat_p_omega) We take the following as an axiom:
L920Definition. We define
finite to be
λX ⇒ ∃n ∈ ω, equip X n of type
set → prop.
L922Axiom. (
nat_finite) We take the following as an axiom:
L924Axiom. (
finite_ind) We take the following as an axiom:
L931Axiom. (
Sing_finite) We take the following as an axiom:
L939Axiom. (
Subq_finite) We take the following as an axiom:
Beginning of Section InfinitePrimes
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_nat.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_nat.
L948Definition. We define
divides_nat to be
λm n ⇒ m ∈ ω ∧ n ∈ ω ∧ ∃k ∈ ω, m * k = n of type
set → set → prop.
End of Section InfinitePrimes
L985Axiom. (
Inj1_eq) We take the following as an axiom:
L987Axiom. (
Inj1I1) We take the following as an axiom:
L989Axiom. (
Inj1I2) We take the following as an axiom:
L991Axiom. (
Inj1E) We take the following as an axiom:
L993Axiom. (
Inj1NE1) We take the following as an axiom:
L995Axiom. (
Inj1NE2) We take the following as an axiom:
L997Definition. We define
Inj0 to be
λX ⇒ {Inj1 x|x ∈ X} of type
set → set.
L1000Axiom. (
Inj0I) We take the following as an axiom:
L1002Axiom. (
Inj0E) We take the following as an axiom:
L1007Axiom. (
Unj_eq) We take the following as an axiom:
L1009Axiom. (
Unj_Inj1_eq) We take the following as an axiom:
L1011Axiom. (
Inj1_inj) We take the following as an axiom:
L1013Axiom. (
Unj_Inj0_eq) We take the following as an axiom:
L1015Axiom. (
Inj0_inj) We take the following as an axiom:
L1017Axiom. (
Inj0_0) We take the following as an axiom:
Notation. We use
+ as an infix operator with priority 450 and which associates to the left corresponding to applying term
setsum.
L1026Axiom. (
Inj0_setsum) We take the following as an axiom:
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
L1028Axiom. (
Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
Beginning of Section pair_setsum
L1045Axiom. (
pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
L1047Axiom. (
pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
L1049Axiom. (
pairE) We take the following as an axiom:
∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)
L1051Axiom. (
pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
L1053Axiom. (
pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
L1055Axiom. (
proj0I) We take the following as an axiom:
L1057Axiom. (
proj0E) We take the following as an axiom:
L1059Axiom. (
proj1I) We take the following as an axiom:
L1061Axiom. (
proj1E) We take the following as an axiom:
L1069Definition. We define
Sigma to be
λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type
set → (set → set) → set.
Notation. We use
∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Sigma.
L1077Axiom. (
proj0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
L1079Axiom. (
proj1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)
L1081Axiom. (
pair_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x
L1083Axiom. (
pair_Sigma_E1) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x
L1085Axiom. (
Sigma_E) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y
L1087Definition. We define
setprod to be
λX Y : set ⇒ ∑x ∈ X, Y of type
set → set → set.
Notation. We use
⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod.
L1091Let lam : set → (set → set) → set ≝ Sigma
L1093Definition. We define
ap to be
λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type
set → set → set.
Notation. When
x is a set, a term
x y is notation for
ap x y.
Notation.
λ x ∈ A ⇒ B is notation for the set
Sigma A (λ x : set ⇒ B).
Notation. We now use n-tuple notation (
a0,...,
an-1) for n ≥ 2 for λ i ∈
n .
if i = 0
then a0 else ... if i =
n-2 then an-2 else an-1.
L1097Axiom. (
lamI) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x
L1099Axiom. (
lamE) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y
L1101Axiom. (
apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
L1103Axiom. (
apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
L1105Axiom. (
beta) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
L1107Axiom. (
proj0_ap_0) We take the following as an axiom:
L1109Axiom. (
proj1_ap_1) We take the following as an axiom:
L1111Axiom. (
pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
L1113Axiom. (
pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
L1115Axiom. (
ap0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
L1117Axiom. (
ap1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))
L1119Definition. We define
pair_p to be
λu : set ⇒ pair (u 0) (u 1) = u of type
set → prop.
L1122Axiom. (
pair_p_I) We take the following as an axiom:
L1126Axiom. (
tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
L1128Definition. We define
Pi to be
λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type
set → (set → set) → set.
Notation. We use
∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Pi.
L1132Axiom. (
PiI) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x
L1135Axiom. (
lam_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)
L1138Axiom. (
ap_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x
L1140Definition. We define
setexp to be
λX Y : set ⇒ ∏y ∈ Y, X of type
set → set → set.
Notation. We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp.
Beginning of Section Tuples
L1148Variable x0 x1 : set
End of Section Tuples
L1155Axiom. (
ReplEq_setprod_ext) We take the following as an axiom:
∀X Y, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y, F x y = G x y) → {F (w 0) (w 1)|w ∈ X ⨯ Y} = {G (w 0) (w 1)|w ∈ X ⨯ Y}
L1157Axiom. (
lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
L1159Axiom. (
tuple_2_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x
L1161Axiom. (
tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
End of Section pair_setsum
Notation. We use
∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Sigma.
Notation. We use
⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod.
Notation. We use
∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Pi.
Notation. We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp.
L1175Definition. We define
DescrR_i_io_1 to be
λR ⇒ Eps_i (λx ⇒ (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) of type
(set → (set → prop) → prop) → set.
L1180Definition. We define
PNoEq_ to be
λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type
set → (set → prop) → (set → prop) → prop.
L1186Axiom. (
PNoEq_ref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
L1188Axiom. (
PNoEq_sym_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
L1190Axiom. (
PNoEq_tra_) We take the following as an axiom:
L1194Definition. We define
PNoLt_ to be
λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type
set → (set → prop) → (set → prop) → prop.
L1197Axiom. (
PNoLt_E_) We take the following as an axiom:
L1200Axiom. (
PNoLt_irref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
L1202Axiom. (
PNoLt_mon_) We take the following as an axiom:
L1214Axiom. (
PNoLtI1) We take the following as an axiom:
L1217Axiom. (
PNoLtI2) We take the following as an axiom:
L1220Axiom. (
PNoLtI3) We take the following as an axiom:
L1223Axiom. (
PNoLtE) We take the following as an axiom:
L1231Axiom. (
PNoLt_irref) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
L1237Axiom. (
PNoLtEq_tra) We take the following as an axiom:
L1239Axiom. (
PNoEqLt_tra) We take the following as an axiom:
L1241Axiom. (
PNoLt_tra) We take the following as an axiom:
L1243Definition. We define
PNoLe to be
λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type
set → (set → prop) → set → (set → prop) → prop.
L1246Axiom. (
PNoLeI1) We take the following as an axiom:
L1249Axiom. (
PNoLeI2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
L1252Axiom. (
PNoLe_ref) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
L1258Axiom. (
PNoLtLe_tra) We take the following as an axiom:
L1260Axiom. (
PNoLeLt_tra) We take the following as an axiom:
L1262Axiom. (
PNoEqLe_tra) We take the following as an axiom:
L1264Axiom. (
PNoLe_tra) We take the following as an axiom:
L1266Definition. We define
PNo_downc to be
λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type
(set → (set → prop) → prop) → set → (set → prop) → prop.
L1269Definition. We define
PNo_upc to be
λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type
(set → (set → prop) → prop) → set → (set → prop) → prop.
L1271Axiom. (
PNoLe_downc) We take the following as an axiom:
L1275Axiom. (
PNo_downc_ref) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p
L1277Axiom. (
PNo_upc_ref) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p
L1279Axiom. (
PNoLe_upc) We take the following as an axiom:
L1283Definition. We define
PNoLt_pwise to be
λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.
L1325Axiom. (
PNo_extend0_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
L1327Axiom. (
PNo_extend1_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
L1335Definition. We define
PNo_lenbdd to be
λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type
set → (set → (set → prop) → prop) → prop.
L1412Definition. We define
PNo_least_rep2 to be
λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
L1444Axiom. (
PNo_bd_pred) We take the following as an axiom:
L1451Axiom. (
PNo_bd_In) We take the following as an axiom:
Beginning of Section TaggedSets
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
L1481Definition. We define
SNoElts_ to be
λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type
set → set.
L1483Axiom. (
SNoElts_mon) We take the following as an axiom:
L1491Axiom. (
PNoEq_PSNo) We take the following as an axiom:
L1493Axiom. (
SNo_PSNo) We take the following as an axiom:
L1499Definition. We define
SNo to be
λx ⇒ ∃alpha, ordinal alpha ∧ SNo_ alpha x of type
set → prop.
L1500Axiom. (
SNo_SNo) We take the following as an axiom:
L1507Axiom. (
SNoLev_uniq) We take the following as an axiom:
L1509Axiom. (
SNoLev_prop) We take the following as an axiom:
L1513Axiom. (
SNoLev_) We take the following as an axiom:
L1517Axiom. (
SNoLev_PSNo) We take the following as an axiom:
L1519Axiom. (
SNo_Subq) We take the following as an axiom:
L1521Definition. We define
SNoEq_ to be
λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type
set → set → set → prop.
L1524Axiom. (
SNoEq_I) We take the following as an axiom:
L1526Axiom. (
SNo_eq) We take the following as an axiom:
End of Section TaggedSets
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
L1539Axiom. (
SNoLtLe) We take the following as an axiom:
L1541Axiom. (
SNoLeE) We take the following as an axiom:
L1543Axiom. (
SNoEq_sym_) We take the following as an axiom:
L1545Axiom. (
SNoEq_tra_) We take the following as an axiom:
L1547Axiom. (
SNoLtE) We take the following as an axiom:
L1554Axiom. (
SNoLtI2) We take the following as an axiom:
L1561Axiom. (
SNoLtI3) We take the following as an axiom:
L1567Axiom. (
SNoLt_irref) We take the following as an axiom:
L1578Axiom. (
SNoLt_tra) We take the following as an axiom:
L1580Axiom. (
SNoLe_ref) We take the following as an axiom:
L1584Axiom. (
SNoLtLe_tra) We take the following as an axiom:
L1586Axiom. (
SNoLeLt_tra) We take the following as an axiom:
L1588Axiom. (
SNoLe_tra) We take the following as an axiom:
L1590Axiom. (
SNoLtLe_or) We take the following as an axiom:
L1603Definition. We define
SNoCutP to be
λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type
set → set → prop.
L1625Axiom. (
SNoCutP_L_0) We take the following as an axiom:
L1627Axiom. (
SNoCutP_0_0) We take the following as an axiom:
L1631Axiom. (
SNoS_E) We take the following as an axiom:
Beginning of Section TaggedSets2
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
L1637Axiom. (
SNoS_I) We take the following as an axiom:
L1639Axiom. (
SNoS_I2) We take the following as an axiom:
L1641Axiom. (
SNoS_Subq) We take the following as an axiom:
L1646Axiom. (
SNoS_E2) We take the following as an axiom:
L1651Axiom. (
SNoS_In_neq) We take the following as an axiom:
L1653Axiom. (
SNoS_SNoLev) We take the following as an axiom:
L1660Axiom. (
SNoL_E) We take the following as an axiom:
L1665Axiom. (
SNoR_E) We take the following as an axiom:
L1670Axiom. (
SNoL_SNoS_) We take the following as an axiom:
L1672Axiom. (
SNoR_SNoS_) We take the following as an axiom:
L1674Axiom. (
SNoL_SNoS) We take the following as an axiom:
L1676Axiom. (
SNoR_SNoS) We take the following as an axiom:
L1678Axiom. (
SNoL_I) We take the following as an axiom:
L1680Axiom. (
SNoR_I) We take the following as an axiom:
L1682Axiom. (
SNo_eta) We take the following as an axiom:
L1697Axiom. (
SNoCut_Le) We take the following as an axiom:
L1702Axiom. (
SNoCut_ext) We take the following as an axiom:
L1733Axiom. (
ordinal_SNo) We take the following as an axiom:
L1743Axiom. (
nat_p_SNo) We take the following as an axiom:
L1745Axiom. (
omega_SNo) We take the following as an axiom:
L1759Axiom. (
SNo_0) We take the following as an axiom:
L1761Axiom. (
SNo_1) We take the following as an axiom:
L1763Axiom. (
SNo_2) We take the following as an axiom:
L1765Axiom. (
SNoLev_0) We take the following as an axiom:
L1767Axiom. (
SNoCut_0_0) We take the following as an axiom:
L1769Axiom. (
SNoL_0) We take the following as an axiom:
L1771Axiom. (
SNoR_0) We take the following as an axiom:
L1773Axiom. (
SNoL_1) We take the following as an axiom:
L1775Axiom. (
SNoR_1) We take the following as an axiom:
End of Section TaggedSets2
L1810Axiom. (
SNo_etaE) We take the following as an axiom:
L1819Axiom. (
SNo_ind) We take the following as an axiom:
∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z
Beginning of Section SurrealRecI
L1830Variable F : set → (set → set) → set
L1845Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecI
Beginning of Section SurrealRecII
L1855Variable F : set → (set → (set → set)) → (set → set)
L1857Let G : set → (set → set → (set → set)) → set → (set → set) ≝ λalpha g ⇒ If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)
L1868Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecII
Beginning of Section SurrealRec2
L1877Variable F : set → set → (set → set → set) → set
L1879Let G : set → (set → set → set) → set → (set → set) → set ≝ λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
L1905Axiom. (
SNo_rec2_eq) We take the following as an axiom:
End of Section SurrealRec2
L1930Axiom. (
SNoLev_ind) We take the following as an axiom:
L1935Axiom. (
SNoLev_ind2) We take the following as an axiom:
L1943Axiom. (
SNoLev_ind3) We take the following as an axiom:
∀P : set → set → set → prop, (∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), P u y z) → (∀v ∈ SNoS_ (SNoLev y), P x v z) → (∀w ∈ SNoS_ (SNoLev z), P x y w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w) → P x y z) → ∀x y z, SNo x → SNo y → SNo z → P x y z
L1955Axiom. (
SNo_omega) We take the following as an axiom:
L1957Axiom. (
SNoLt_0_1) We take the following as an axiom:
L1959Axiom. (
SNoLt_0_2) We take the following as an axiom:
L1961Axiom. (
SNoLt_1_2) We take the following as an axiom:
Beginning of Section SurrealMinus
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
L2027Axiom. (
add_SNo_eq) We take the following as an axiom:
L2038Axiom. (
SNo_add_SNo) We take the following as an axiom:
L2044Axiom. (
add_SNo_Lt1) We take the following as an axiom:
L2046Axiom. (
add_SNo_Le1) We take the following as an axiom:
L2048Axiom. (
add_SNo_Lt2) We take the following as an axiom:
L2050Axiom. (
add_SNo_Le2) We take the following as an axiom:
L2056Axiom. (
add_SNo_Lt3) We take the following as an axiom:
L2058Axiom. (
add_SNo_Le3) We take the following as an axiom:
L2062Axiom. (
add_SNo_com) We take the following as an axiom:
L2064Axiom. (
add_SNo_0L) We take the following as an axiom:
L2066Axiom. (
add_SNo_0R) We take the following as an axiom:
L2110Axiom. (
minus_SNo_0) We take the following as an axiom:
End of Section SurrealAdd
Beginning of Section SurrealMul
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
L2177Axiom. (
mul_SNo_eq) We take the following as an axiom:
L2186Axiom. (
mul_SNo_eq_2) We take the following as an axiom:
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
L2207Axiom. (
mul_SNo_prop_1) We take the following as an axiom:
∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p
L2217Axiom. (
SNo_mul_SNo) We take the following as an axiom:
L2221Axiom. (
mul_SNo_Lt) We take the following as an axiom:
L2224Axiom. (
mul_SNo_Le) We take the following as an axiom:
End of Section SurrealMul
Beginning of Section Int
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
L2263Definition. We define
int to be
ω ∪ {- n|n ∈ ω} of type
set.
L2265Axiom. (
int_SNo_cases) We take the following as an axiom:
∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x
L2270Axiom. (
int_3_cases) We take the following as an axiom:
L2276Axiom. (
int_SNo) We take the following as an axiom:
L2284Axiom. (
int_add_SNo) We take the following as an axiom:
L2288Axiom. (
int_mul_SNo) We take the following as an axiom:
End of Section Int
Beginning of Section SurrealExp
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
End of Section SurrealExp