Primitive. The name Eps_i is a term of type (set → prop) → set.
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
In Proofgold the corresponding term root is c3f0de... and proposition id is 756d6e...
Definition. We define True to be ∀p : prop, p → p of type prop.
In Proofgold the corresponding term root is f81b39... and object id is 94e6a7...
Definition. We define False to be ∀p : prop, p of type prop.
In Proofgold the corresponding term root is 1db705... and object id is 266ad5...
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
In Proofgold the corresponding term root is f30435... and object id is 0dc3c7...
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 2ba7d0... and object id is 37da83...
Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 957746... and object id is 86ae64...
Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.
In Proofgold the corresponding term root is 98aaaf... and object id is 6588e5...
Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
Variable A : SType
Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.
Definition. 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
Variable A B : SType
Axiom. (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
Variable A : SType
Definition. 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.
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
In Proofgold the corresponding term root is 00ac81... and proposition id is 1d1e56...
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.
Definition. We define Subq to be λA B ⇒ ∀x ∈ A, x ∈ B of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
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.
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
In Proofgold the corresponding term root is b3b698... and proposition id is 21238a...
Axiom. (In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
In Proofgold the corresponding term root is 045211... and proposition id is 20faa5...
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.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x ∈ Empty
In Proofgold the corresponding term root is 4da931... and proposition id is c7b0d7...
Primitive. The name ⋃ is a term of type set → set.
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is bf4c26... and proposition id is 40660f...
Primitive. The name 𝒫 is a term of type set → set.
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
In Proofgold the corresponding term root is 746e5a... and proposition id is d67d71...
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).
Axiom. (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
In Proofgold the corresponding term root is 4f2357... and proposition id is 471122...
Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.
In Proofgold the corresponding term root is 538bb7... and object id is e69f92...
Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.
In Proofgold the corresponding term root is 57561d... and object id is e45ed9...
Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.
In Proofgold the corresponding term root is 8b9cc0... and object id is d2954e...
Definition. 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.
In Proofgold the corresponding term root is 5574bb... and object id is 513534...
Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.
In Proofgold the corresponding term root is 1bd4aa... and object id is e1eccf...
Primitive. The name UnivOf is a term of type set → set.
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N ∈ UnivOf N
In Proofgold the corresponding term root is efcc16... and proposition id is e805d4...
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
In Proofgold the corresponding term root is 053575... and proposition id is cf4f5c...
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
In Proofgold the corresponding term root is 770e35... and proposition id is 0e7363...
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U
In Proofgold the corresponding term root is 042dda... and proposition id is 1f7933...
Axiom. (FalseE) We take the following as an axiom:
False → ∀p : prop, p
In Proofgold the corresponding term root is 0dcfe3... and proposition id is f9a230...
Axiom. (TrueI) We take the following as an axiom:
True
In Proofgold the corresponding term root is f81b39... and proposition id is eb5cbf...
Axiom. (andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
In Proofgold the corresponding term root is 3c4f59... and proposition id is 3bf198...
Axiom. (andEL) We take the following as an axiom:
∀A B : prop, A ∧ B → A
In Proofgold the corresponding term root is c2f006... and proposition id is 718be4...
Axiom. (andER) We take the following as an axiom:
∀A B : prop, A ∧ B → B
In Proofgold the corresponding term root is f66b82... and proposition id is 5caea9...
Axiom. (orIL) We take the following as an axiom:
∀A B : prop, A → A ∨ B
In Proofgold the corresponding term root is 55be8c... and proposition id is 382673...
Axiom. (orIR) We take the following as an axiom:
∀A B : prop, B → A ∨ B
In Proofgold the corresponding term root is 3c63fc... and proposition id is 8d8e40...
Beginning of Section PropN
Variable P1 P2 P3 : prop
Axiom. (and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
In Proofgold the corresponding term root is 23ac9c... and proposition id is cb999b...
Axiom. (and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
In Proofgold the corresponding term root is 445f76... and proposition id is 50fabe...
Axiom. (or3I1) We take the following as an axiom:
P1 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is e44f34... and proposition id is ef9071...
Axiom. (or3I2) We take the following as an axiom:
P2 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 8df6e1... and proposition id is 827a91...
Axiom. (or3I3) We take the following as an axiom:
P3 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 017a00... and proposition id is 3b7f5a...
Axiom. (or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
In Proofgold the corresponding term root is 5d069d... and proposition id is 2120c9...
Variable P4 : prop
Axiom. (and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
In Proofgold the corresponding term root is ed32cb... and proposition id is 906aaf...
Variable P5 : prop
Axiom. (and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
In Proofgold the corresponding term root is 041ced... and proposition id is 11fc47...
End of Section PropN
Axiom. (not_or_and_demorgan) We take the following as an axiom:
∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B
In Proofgold the corresponding term root is 8733b0... and proposition id is 27db91...
Axiom. (not_ex_all_demorgan_i) We take the following as an axiom:
∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x
In Proofgold the corresponding term root is 73523e... and proposition id is 075ddf...
Axiom. (iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
In Proofgold the corresponding term root is 7516b9... and proposition id is 4d966a...
Axiom. (iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
In Proofgold the corresponding term root is 0e94c6... and proposition id is 9a07f6...
Axiom. (iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
In Proofgold the corresponding term root is 54702a... and proposition id is f5dfc0...
Axiom. (iff_refl) We take the following as an axiom:
∀A : prop, A ↔ A
In Proofgold the corresponding term root is 01338d... and proposition id is 7b71b5...
Axiom. (iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
In Proofgold the corresponding term root is 00bac1... and proposition id is 11f51b...
Axiom. (iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
In Proofgold the corresponding term root is 363285... and proposition id is c3b8f1...
Axiom. (eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
In Proofgold the corresponding term root is 193bbf... and proposition id is 7ecbc4...
Axiom. (f_eq_i) We take the following as an axiom:
∀f : set → set, ∀x y, x = y → f x = f y
In Proofgold the corresponding term root is b0871b... and proposition id is 2a60fd...
Axiom. (neq_i_sym) We take the following as an axiom:
∀x y, x ≠ y → y ≠ x
In Proofgold the corresponding term root is 5c0ec9... and proposition id is 1a9898...
Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.
In Proofgold the corresponding term root is 36808c... and object id is 253451...
Notation. We use ∉ as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
Axiom. (Eps_i_ex) We take the following as an axiom:
∀P : set → prop, (∃x, P x) → P (Eps_i P)
In Proofgold the corresponding term root is be36f0... and proposition id is 6a2832...
Axiom. (pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
In Proofgold the corresponding term root is e9e64b... and proposition id is d4b2d3...
Axiom. (prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
In Proofgold the corresponding term root is a9ede2... and proposition id is a56fc7...
Axiom. (Subq_ref) We take the following as an axiom:
∀X : set, X ⊆ X
In Proofgold the corresponding term root is 4c5395... and proposition id is 2efbac...
Axiom. (Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
In Proofgold the corresponding term root is 79accb... and proposition id is fed662...
Axiom. (Subq_contra) We take the following as an axiom:
∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X
In Proofgold the corresponding term root is d15e87... and proposition id is bd5a2f...
Axiom. (EmptyE) We take the following as an axiom:
∀x : set, x ∉ Empty
In Proofgold the corresponding term root is 93a77a... and proposition id is 8349ab...
Axiom. (Subq_Empty) We take the following as an axiom:
∀X : set, Empty ⊆ X
In Proofgold the corresponding term root is a517cd... and proposition id is 2faec3...
Axiom. (Empty_Subq_eq) We take the following as an axiom:
∀X : set, X ⊆ Empty → X = Empty
In Proofgold the corresponding term root is 2dd959... and proposition id is 988691...
Axiom. (Empty_eq) We take the following as an axiom:
∀X : set, (∀x, x ∉ X) → X = Empty
In Proofgold the corresponding term root is 3de0fd... and proposition id is 8b915b...
Axiom. (UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
In Proofgold the corresponding term root is 51b549... and proposition id is bade05...
Axiom. (UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is db4bc1... and proposition id is ab781b...
Axiom. (UnionE_impred) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
In Proofgold the corresponding term root is 8efd57... and proposition id is d621b2...
Axiom. (PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...
Axiom. (PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
In Proofgold the corresponding term root is 149030... and proposition id is 647070...
Axiom. (Empty_In_Power) We take the following as an axiom:
∀X : set, Empty ∈ 𝒫 X
In Proofgold the corresponding term root is d2656b... and proposition id is 487855...
Axiom. (Self_In_Power) We take the following as an axiom:
∀X : set, X ∈ 𝒫 X
In Proofgold the corresponding term root is f81ef2... and proposition id is 8336ff...
Axiom. (xm) We take the following as an axiom:
∀P : prop, P ∨ ¬ P
In Proofgold the corresponding term root is 62e4c3... and proposition id is 7b6737...
Axiom. (dneg) We take the following as an axiom:
∀P : prop, ¬ ¬ P → P
In Proofgold the corresponding term root is 852f34... and proposition id is b7cc63...
Axiom. (not_all_ex_demorgan_i) We take the following as an axiom:
∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x
In Proofgold the corresponding term root is 9bfa06... and proposition id is 4b1474...
Axiom. (eq_or_nand) We take the following as an axiom:
or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))
In Proofgold the corresponding term root is 24ac29... and proposition id is d212ef...
Primitive. The name exactly1of2 is a term of type prop → prop → prop.
In Proofgold the corresponding term root is 163602... and object id is a0b944...
Axiom. (exactly1of2_I1) We take the following as an axiom:
∀A B : prop, A → ¬ B → exactly1of2 A B
In Proofgold the corresponding term root is 8e1e45... and proposition id is 4f94fa...
Axiom. (exactly1of2_I2) We take the following as an axiom:
∀A B : prop, ¬ A → B → exactly1of2 A B
In Proofgold the corresponding term root is c9fd90... and proposition id is 6c1ad9...
Axiom. (exactly1of2_E) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
In Proofgold the corresponding term root is 29b901... and proposition id is c67434...
Axiom. (exactly1of2_or) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → A ∨ B
In Proofgold the corresponding term root is 18af73... and proposition id is e092f6...
Axiom. (ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
In Proofgold the corresponding term root is b146c5... and proposition id is f84982...
Axiom. (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
In Proofgold the corresponding term root is 311cdf... and proposition id is d07e78...
Axiom. (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
In Proofgold the corresponding term root is 31e90a... and proposition id is 34fefd...
Axiom. (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
In Proofgold the corresponding term root is e374fe... and proposition id is d4b7b0...
Axiom. (Repl_Empty) We take the following as an axiom:
∀F : set → set, {F x|x ∈ Empty} = Empty
In Proofgold the corresponding term root is 2af2ac... and proposition id is 0835bc...
Axiom. (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}
In Proofgold the corresponding term root is eb4c1d... and proposition id is 56aad0...
Axiom. (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}
In Proofgold the corresponding term root is 81eb90... and proposition id is 781846...
Axiom. (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
In Proofgold the corresponding term root is 968c7a... and proposition id is 9ce73d...
Axiom. (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
In Proofgold the corresponding term root is e15a6b... and proposition id is 5faff0...
Primitive. The name If_i is a term of type prop → set → set → set.
In Proofgold the corresponding term root is b8ff52... and object id is 76c4c0...
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.
Axiom. (If_i_correct) We take the following as an axiom:
∀p : prop, ∀x y : set, p ∧ (if p then x else y) = x ∨ ¬ p ∧ (if p then x else y) = y
In Proofgold the corresponding term root is 2a17ae... and proposition id is b5c26b...
Axiom. (If_i_0) We take the following as an axiom:
∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y
In Proofgold the corresponding term root is fde1a8... and proposition id is a9abd3...
Axiom. (If_i_1) We take the following as an axiom:
∀p : prop, ∀x y : set, p → (if p then x else y) = x
In Proofgold the corresponding term root is b1aa3f... and proposition id is a93c2c...
Axiom. (If_i_or) We take the following as an axiom:
∀p : prop, ∀x y : set, (if p then x else y) = x ∨ (if p then x else y) = y
In Proofgold the corresponding term root is 3e1439... and proposition id is 4de742...
Primitive. The name UPair is a term of type set → set → set.
In Proofgold the corresponding term root is 742438... and object id is dc3134...
Notation. {x,y} is notation for UPair x y.
Axiom. (UPairE) We take the following as an axiom:
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
In Proofgold the corresponding term root is 5009cc... and proposition id is e67fc5...
Axiom. (UPairI1) We take the following as an axiom:
∀y z : set, y ∈ {y,z}
In Proofgold the corresponding term root is b1eea2... and proposition id is bf3649...
Axiom. (UPairI2) We take the following as an axiom:
∀y z : set, z ∈ {y,z}
In Proofgold the corresponding term root is 27455b... and proposition id is a0a5b8...
Primitive. The name Sing is a term of type set → set.
In Proofgold the corresponding term root is bd01a8... and object id is b7dfc1...
Notation. {x} is notation for Sing x.
Axiom. (SingI) We take the following as an axiom:
∀x : set, x ∈ {x}
In Proofgold the corresponding term root is 6591bd... and proposition id is 2f93be...
Axiom. (SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
In Proofgold the corresponding term root is 9341fe... and proposition id is 3b0dd8...
Axiom. (Sing_inj) We take the following as an axiom:
∀x y, {x} = {y} → x = y
In Proofgold the corresponding term root is dea1af... and proposition id is 96eb72...
Primitive. The name binunion is a term of type set → set → set.
In Proofgold the corresponding term root is 5e1ac4... and object id is e14989...
Notation. We use ∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
Axiom. (binunionI1) We take the following as an axiom:
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
In Proofgold the corresponding term root is 753b6e... and proposition id is e43c8d...
Axiom. (binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
In Proofgold the corresponding term root is b8821c... and proposition id is 959535...
Axiom. (binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
In Proofgold the corresponding term root is 5a3195... and proposition id is 1ceeb0...
Axiom. (binunionE') We take the following as an axiom:
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
In Proofgold the corresponding term root is e5b951... and proposition id is 570dd1...
Axiom. (binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
In Proofgold the corresponding term root is e0f701... and proposition id is 61dfba...
Axiom. (binunion_com_Subq) We take the following as an axiom:
∀X Y : set, X ∪ Y ⊆ Y ∪ X
In Proofgold the corresponding term root is ee8f5a... and proposition id is d84076...
Axiom. (binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
In Proofgold the corresponding term root is e20f4f... and proposition id is bed33e...
Axiom. (binunion_idl) We take the following as an axiom:
∀X : set, Empty ∪ X = X
In Proofgold the corresponding term root is 8dc616... and proposition id is 52cd24...
Axiom. (binunion_idr) We take the following as an axiom:
∀X : set, X ∪ Empty = X
In Proofgold the corresponding term root is 67e82f... and proposition id is d5a005...
Axiom. (binunion_Subq_1) We take the following as an axiom:
∀X Y : set, X ⊆ X ∪ Y
In Proofgold the corresponding term root is b2c658... and proposition id is a9b3a8...
Axiom. (binunion_Subq_2) We take the following as an axiom:
∀X Y : set, Y ⊆ X ∪ Y
In Proofgold the corresponding term root is 3cf229... and proposition id is 1507ce...
Axiom. (binunion_Subq_min) We take the following as an axiom:
∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z
In Proofgold the corresponding term root is cc32a5... and proposition id is d64da4...
Axiom. (Subq_binunion_eq) We take the following as an axiom:
∀X Y, (X ⊆ Y) = (X ∪ Y = Y)
In Proofgold the corresponding term root is f8ee53... and proposition id is c2d393...
Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.
In Proofgold the corresponding term root is 1f3a09... and object id is 5bdda8...
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.
Primitive. The name famunion is a term of type set → (set → set) → set.
In Proofgold the corresponding term root is b3e3bf... and object id is 241349...
Notation. We use ⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
Axiom. (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
In Proofgold the corresponding term root is 0c23d0... and proposition id is e4f490...
Axiom. (famunionE) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
In Proofgold the corresponding term root is 48683a... and proposition id is ede6e5...
Axiom. (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
In Proofgold the corresponding term root is ab2266... and proposition id is 562023...
Axiom. (famunion_Empty) We take the following as an axiom:
∀F : set → set, (⋃x ∈ EmptyF x) = Empty
In Proofgold the corresponding term root is fc1bdd... and proposition id is 22bd10...
Axiom. (famunion_Subq) We take the following as an axiom:
∀X, ∀f g : set → set, (∀x ∈ X, f x ⊆ g x) → famunion X f ⊆ famunion X g
In Proofgold the corresponding term root is 5f0b79... and proposition id is b1b158...
Axiom. (famunion_ext) We take the following as an axiom:
∀X, ∀f g : set → set, (∀x ∈ X, f x = g x) → famunion X f = famunion X g
In Proofgold the corresponding term root is 8dc4c3... and proposition id is 5bc238...
Beginning of Section SepSec
Variable X : set
Variable P : set → prop
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
Let F : set → set ≝ λx ⇒ if P x then x else z
Primitive. The name Sep is a term of type set.
In Proofgold the corresponding term root is f336a4... and object id is 7008ec...
End of Section SepSec
Notation. {x ∈ A | B} is notation for Sep A (λ x . B).
Axiom. (SepI) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
In Proofgold the corresponding term root is ea56bb... and proposition id is 147a6a...
Axiom. (SepE) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
In Proofgold the corresponding term root is 102830... and proposition id is 00899a...
Axiom. (SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
In Proofgold the corresponding term root is d83445... and proposition id is 6399b0...
Axiom. (SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
In Proofgold the corresponding term root is aa61ca... and proposition id is 104227...
Axiom. (Sep_Empty) We take the following as an axiom:
∀P : set → prop, {x ∈ Empty|P x} = Empty
In Proofgold the corresponding term root is f36949... and proposition id is e3473a...
Axiom. (Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
In Proofgold the corresponding term root is 97d080... and proposition id is c17d45...
Axiom. (Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
In Proofgold the corresponding term root is 612c94... and proposition id is fba53e...
Primitive. The name ReplSep is a term of type set → (set → prop) → (set → set) → set.
In Proofgold the corresponding term root is ec807b... and object id is 9bdb56...
Notation. {B| x ∈ A, C} is notation for ReplSep A (λ x . C) (λ x . B).
Axiom. (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}
In Proofgold the corresponding term root is 35b8ec... and proposition id is aa7b2b...
Axiom. (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
In Proofgold the corresponding term root is 415ff1... and proposition id is b1bc76...
Axiom. (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
In Proofgold the corresponding term root is 5d26e1... and proposition id is 063cea...
Primitive. The name binintersect is a term of type set → set → set.
In Proofgold the corresponding term root is b2abd2... and object id is 2c68d8...
Notation. We use ∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
Axiom. (binintersectI) We take the following as an axiom:
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
In Proofgold the corresponding term root is fd656d... and proposition id is 9081a0...
Axiom. (binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
In Proofgold the corresponding term root is 463b5a... and proposition id is ca0636...
Axiom. (binintersectE1) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X
In Proofgold the corresponding term root is b12123... and proposition id is 51005e...
Axiom. (binintersectE2) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ Y
In Proofgold the corresponding term root is 39656f... and proposition id is 599de4...
Axiom. (binintersect_Subq_1) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ X
In Proofgold the corresponding term root is f296ad... and proposition id is 303833...
Axiom. (binintersect_Subq_2) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y
In Proofgold the corresponding term root is fcd5f5... and proposition id is 8b9fd2...
Axiom. (binintersect_Subq_eq_1) We take the following as an axiom:
∀X Y, X ⊆ Y → X ∩ Y = X
In Proofgold the corresponding term root is 277eec... and proposition id is ec1cc3...
Axiom. (binintersect_Subq_max) We take the following as an axiom:
∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y
In Proofgold the corresponding term root is 95624d... and proposition id is 64636b...
Axiom. (binintersect_com_Subq) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y ∩ X
In Proofgold the corresponding term root is 410255... and proposition id is 5977b8...
Axiom. (binintersect_com) We take the following as an axiom:
∀X Y : set, X ∩ Y = Y ∩ X
In Proofgold the corresponding term root is 1a2c6f... and proposition id is ea8a1c...
Primitive. The name setminus is a term of type set → set → set.
In Proofgold the corresponding term root is c68e5a... and object id is 913da1...
Notation. We use ∖ as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
Axiom. (setminusI) We take the following as an axiom:
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
In Proofgold the corresponding term root is 0feeb8... and proposition id is 80db6a...
Axiom. (setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
In Proofgold the corresponding term root is 6774f4... and proposition id is 71763a...
Axiom. (setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
In Proofgold the corresponding term root is 064bd5... and proposition id is d74baf...
Axiom. (setminusE2) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∉ Y
In Proofgold the corresponding term root is 3e6969... and proposition id is 36b273...
Axiom. (setminus_Subq) We take the following as an axiom:
∀X Y : set, X ∖ Y ⊆ X
In Proofgold the corresponding term root is 5a497a... and proposition id is 66f982...
Axiom. (setminus_Subq_contra) We take the following as an axiom:
∀X Y Z : set, Z ⊆ Y → X ∖ Y ⊆ X ∖ Z
In Proofgold the corresponding term root is b1f440... and proposition id is 9a06b2...
Axiom. (setminus_In_Power) We take the following as an axiom:
∀A U, A ∖ U ∈ 𝒫 A
In Proofgold the corresponding term root is 062275... and proposition id is f503e1...
Axiom. (setminus_idr) We take the following as an axiom:
∀X, X ∖ Empty = X
In Proofgold the corresponding term root is 20e108... and proposition id is ebb8ec...
Axiom. (In_irref) We take the following as an axiom:
∀x, x ∉ x
In Proofgold the corresponding term root is 12629a... and proposition id is 7bfd46...
Axiom. (In_no2cycle) We take the following as an axiom:
∀x y, x ∈ y → y ∈ x → False
In Proofgold the corresponding term root is 6d0cb3... and proposition id is 344f30...
Primitive. The name ordsucc is a term of type set → set.
In Proofgold the corresponding term root is 65d883... and object id is 1452f7...
Axiom. (ordsuccI1) We take the following as an axiom:
∀x : set, x ⊆ ordsucc x
In Proofgold the corresponding term root is 5a2f15... and proposition id is 367d80...
Axiom. (ordsuccI2) We take the following as an axiom:
∀x : set, x ∈ ordsucc x
In Proofgold the corresponding term root is 2f6efd... and proposition id is 2be6be...
Axiom. (ordsuccE) We take the following as an axiom:
∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x
In Proofgold the corresponding term root is 01c4ee... and proposition id is 3849f5...
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
Axiom. (neq_0_ordsucc) We take the following as an axiom:
∀a : set, 0 ≠ ordsucc a
In Proofgold the corresponding term root is 1b20cb... and proposition id is 04c410...
Axiom. (neq_ordsucc_0) We take the following as an axiom:
∀a : set, ordsucc a ≠ 0
In Proofgold the corresponding term root is 9b391e... and proposition id is 346ee0...
Axiom. (ordsucc_inj) We take the following as an axiom:
∀a b : set, ordsucc a = ordsucc b → a = b
In Proofgold the corresponding term root is 40cad6... and proposition id is 4869a1...
Axiom. (ordsucc_inj_contra) We take the following as an axiom:
∀a b : set, a ≠ b → ordsucc a ≠ ordsucc b
In Proofgold the corresponding term root is 791304... and proposition id is 66f4df...
Axiom. (In_0_1) We take the following as an axiom:
0 ∈ 1
In Proofgold the corresponding term root is e984b0... and proposition id is 3be80a...
Axiom. (In_0_2) We take the following as an axiom:
0 ∈ 2
In Proofgold the corresponding term root is 56dc2c... and proposition id is 5a202e...
Axiom. (In_1_2) We take the following as an axiom:
1 ∈ 2
In Proofgold the corresponding term root is 8704b3... and proposition id is 431e9e...
Axiom. (In_1_3) We take the following as an axiom:
1 ∈ 3
In Proofgold the corresponding term root is 85201c... and proposition id is 8f3cec...
Axiom. (In_2_3) We take the following as an axiom:
2 ∈ 3
In Proofgold the corresponding term root is 4d67ce... and proposition id is ebdb5d...
Axiom. (In_1_4) We take the following as an axiom:
1 ∈ 4
In Proofgold the corresponding term root is 5cabbb... and proposition id is 1e2d5b...
Axiom. (In_2_4) We take the following as an axiom:
2 ∈ 4
In Proofgold the corresponding term root is 33c2ff... and proposition id is 2cbb94...
Axiom. (In_3_4) We take the following as an axiom:
3 ∈ 4
In Proofgold the corresponding term root is f25e44... and proposition id is 22a0af...
Axiom. (In_1_5) We take the following as an axiom:
1 ∈ 5
In Proofgold the corresponding term root is be96a3... and proposition id is 34c02d...
Axiom. (In_2_5) We take the following as an axiom:
2 ∈ 5
In Proofgold the corresponding term root is ec68b2... and proposition id is 7ed8e4...
Axiom. (In_3_5) We take the following as an axiom:
3 ∈ 5
In Proofgold the corresponding term root is eddd36... and proposition id is df92fe...
Axiom. (In_4_5) We take the following as an axiom:
4 ∈ 5
In Proofgold the corresponding term root is e72ad7... and proposition id is 03b9ff...
Axiom. (In_1_6) We take the following as an axiom:
1 ∈ 6
In Proofgold the corresponding term root is 4c8428... and proposition id is 327244...
Axiom. (In_1_7) We take the following as an axiom:
1 ∈ 7
In Proofgold the corresponding term root is aa5cde... and proposition id is 392ce8...
Axiom. (In_1_8) We take the following as an axiom:
1 ∈ 8
In Proofgold the corresponding term root is fbef0f... and proposition id is 0d7bd5...
Definition. 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.
In Proofgold the corresponding term root is 458be3... and object id is cfd8eb...
Axiom. (nat_0) We take the following as an axiom:
nat_p 0
In Proofgold the corresponding term root is 1617bc... and proposition id is 457a72...
Axiom. (nat_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → nat_p (ordsucc n)
In Proofgold the corresponding term root is 87f001... and proposition id is 57e940...
Axiom. (nat_1) We take the following as an axiom:
nat_p 1
In Proofgold the corresponding term root is 2f83fa... and proposition id is 672d0d...
Axiom. (nat_2) We take the following as an axiom:
nat_p 2
In Proofgold the corresponding term root is a8be15... and proposition id is 9112e9...
Axiom. (nat_3) We take the following as an axiom:
nat_p 3
In Proofgold the corresponding term root is c0849f... and proposition id is 133bb7...
Axiom. (nat_4) We take the following as an axiom:
nat_p 4
In Proofgold the corresponding term root is 2c8e1a... and proposition id is 69ffc3...
Axiom. (nat_5) We take the following as an axiom:
nat_p 5
In Proofgold the corresponding term root is 1a2477... and proposition id is b84b25...
Axiom. (nat_6) We take the following as an axiom:
nat_p 6
In Proofgold the corresponding term root is 25de88... and proposition id is b4ec67...
Axiom. (nat_7) We take the following as an axiom:
nat_p 7
In Proofgold the corresponding term root is c2e74e... and proposition id is 86ad85...
Axiom. (nat_8) We take the following as an axiom:
nat_p 8
In Proofgold the corresponding term root is 18883c... and proposition id is f93125...
Axiom. (nat_0_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → 0 ∈ ordsucc n
In Proofgold the corresponding term root is 9b50b9... and proposition id is 251f3d...
Axiom. (nat_ordsucc_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n
In Proofgold the corresponding term root is 819399... and proposition id is 4c2c95...
Axiom. (nat_ind) We take the following as an axiom:
∀p : set → prop, p 0 → (∀n, nat_p n → p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is 5d06f5... and proposition id is ecdce8...
Axiom. (nat_inv_impred) We take the following as an axiom:
∀p : set → prop, p 0 → (∀n, nat_p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is eddc5f... and proposition id is 175e22...
Axiom. (nat_inv) We take the following as an axiom:
∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x
In Proofgold the corresponding term root is 5d4296... and proposition id is 4f4167...
Axiom. (nat_complete_ind) We take the following as an axiom:
∀p : set → prop, (∀n, nat_p n → (∀m ∈ n, p m) → p n) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is d32e5b... and proposition id is 1e1608...
Axiom. (nat_p_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, nat_p m
In Proofgold the corresponding term root is e88449... and proposition id is 03b874...
Axiom. (nat_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, m ⊆ n
In Proofgold the corresponding term root is 6bc6e1... and proposition id is cad6fe...
Axiom. (nat_ordsucc_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n
In Proofgold the corresponding term root is e42e58... and proposition id is bc868d...
Axiom. (Union_ordsucc_eq) We take the following as an axiom:
∀n, nat_p n → ⋃ (ordsucc n) = n
In Proofgold the corresponding term root is 87308c... and proposition id is e8cdeb...
Axiom. (cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
In Proofgold the corresponding term root is 706605... and proposition id is c6f0c0...
Axiom. (cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
In Proofgold the corresponding term root is 4693b2... and proposition id is 74563a...
Axiom. (cases_3) We take the following as an axiom:
∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i
In Proofgold the corresponding term root is 2a0ca2... and proposition id is 1a8a20...
Axiom. (neq_0_1) We take the following as an axiom:
0 ≠ 1
In Proofgold the corresponding term root is 9630fd... and proposition id is 74867a...
Axiom. (neq_1_0) We take the following as an axiom:
1 ≠ 0
In Proofgold the corresponding term root is 3a0478... and proposition id is 952fed...
Axiom. (neq_0_2) We take the following as an axiom:
0 ≠ 2
In Proofgold the corresponding term root is 055580... and proposition id is 9c9a3f...
Axiom. (neq_2_0) We take the following as an axiom:
2 ≠ 0
In Proofgold the corresponding term root is 4d54d8... and proposition id is 00b357...
Axiom. (neq_1_2) We take the following as an axiom:
1 ≠ 2
In Proofgold the corresponding term root is 1d088a... and proposition id is e5542e...
Axiom. (neq_1_3) We take the following as an axiom:
1 ≠ 3
In Proofgold the corresponding term root is d3b293... and proposition id is c0a1f7...
Axiom. (neq_2_3) We take the following as an axiom:
2 ≠ 3
In Proofgold the corresponding term root is 9a904f... and proposition id is 2cd643...
Axiom. (neq_2_4) We take the following as an axiom:
2 ≠ 4
In Proofgold the corresponding term root is 0bb647... and proposition id is c4e5df...
Axiom. (neq_3_4) We take the following as an axiom:
3 ≠ 4
In Proofgold the corresponding term root is 031ec9... and proposition id is 66a378...
Axiom. (ZF_closed_E) We take the following as an axiom:
∀U, ZF_closed U → ∀p : prop, (Union_closed U → Power_closed U → Repl_closed U → p) → p
In Proofgold the corresponding term root is d795d4... and proposition id is c78c79...
Axiom. (ZF_Union_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U
In Proofgold the corresponding term root is a7cdf1... and proposition id is 250496...
Axiom. (ZF_Power_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U
In Proofgold the corresponding term root is 5d781f... and proposition id is 7411dd...
Axiom. (ZF_Repl_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → {F x|x ∈ X} ∈ U
In Proofgold the corresponding term root is 312268... and proposition id is 18f8ac...
Axiom. (ZF_UPair_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U
In Proofgold the corresponding term root is 54256a... and proposition id is ef42d3...
Axiom. (ZF_Sing_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U
In Proofgold the corresponding term root is 5d8a83... and proposition id is d02d66...
Axiom. (ZF_binunion_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U
In Proofgold the corresponding term root is 8a6240... and proposition id is 1bacfd...
Axiom. (ZF_ordsucc_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U
In Proofgold the corresponding term root is 68a6ba... and proposition id is 946e61...
Axiom. (nat_p_UnivOf_Empty) We take the following as an axiom:
∀n : set, nat_p n → n ∈ UnivOf Empty
In Proofgold the corresponding term root is c885b4... and proposition id is 645aee...
Primitive. The name ω is a term of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Axiom. (omega_nat_p) We take the following as an axiom:
∀n ∈ ω, nat_p n
In Proofgold the corresponding term root is f44a30... and proposition id is c2c61b...
Axiom. (nat_p_omega) We take the following as an axiom:
∀n : set, nat_p n → n ∈ ω
In Proofgold the corresponding term root is 6a1b44... and proposition id is d5080f...
Axiom. (omega_ordsucc) We take the following as an axiom:
∀n ∈ ω, ordsucc n ∈ ω
In Proofgold the corresponding term root is 246d63... and proposition id is 17fb64...
Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.
In Proofgold the corresponding term root is ee28d9... and object id is 53ee8f...
Axiom. (ordinal_TransSet) We take the following as an axiom:
∀alpha : set, ordinal alpha → TransSet alpha
In Proofgold the corresponding term root is 0dbaa3... and proposition id is c9e708...
Axiom. (ordinal_Empty) We take the following as an axiom:
ordinal Empty
In Proofgold the corresponding term root is ce6bea... and proposition id is 7d95d1...
Axiom. (ordinal_Hered) We take the following as an axiom:
∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta
In Proofgold the corresponding term root is a7e78b... and proposition id is 43b0f3...
Axiom. (TransSet_ordsucc) We take the following as an axiom:
∀X : set, TransSet X → TransSet (ordsucc X)
In Proofgold the corresponding term root is fc2157... and proposition id is ebf9e3...
Axiom. (ordinal_ordsucc) We take the following as an axiom:
∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)
In Proofgold the corresponding term root is 0ba97f... and proposition id is ab0f7f...
Axiom. (nat_p_ordinal) We take the following as an axiom:
∀n : set, nat_p n → ordinal n
In Proofgold the corresponding term root is 274fc4... and proposition id is 47cd49...
Axiom. (ordinal_1) We take the following as an axiom:
ordinal 1
In Proofgold the corresponding term root is c94d41... and proposition id is e325a5...
Axiom. (ordinal_2) We take the following as an axiom:
ordinal 2
In Proofgold the corresponding term root is a57075... and proposition id is e64a0d...
Axiom. (omega_TransSet) We take the following as an axiom:
TransSet ω
In Proofgold the corresponding term root is 8e89e6... and proposition id is c79e75...
Axiom. (omega_ordinal) We take the following as an axiom:
ordinal ω
In Proofgold the corresponding term root is 3d621a... and proposition id is 177484...
Axiom. (ordsucc_omega_ordinal) We take the following as an axiom:
ordinal (ordsucc ω)
In Proofgold the corresponding term root is 66659c... and proposition id is ecfad7...
Axiom. (TransSet_ordsucc_In_Subq) We take the following as an axiom:
∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X
In Proofgold the corresponding term root is d5d882... and proposition id is 3c2e45...
Axiom. (ordinal_ordsucc_In_Subq) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha
In Proofgold the corresponding term root is 57345e... and proposition id is b00304...
Axiom. (ordinal_trichotomy_or) We take the following as an axiom:
∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha
In Proofgold the corresponding term root is 078a34... and proposition id is 21f2d6...
Axiom. (ordinal_trichotomy_or_impred) We take the following as an axiom:
∀alpha beta : set, ordinal alpha → ordinal beta → ∀p : prop, (alpha ∈ beta → p) → (alpha = beta → p) → (beta ∈ alpha → p) → p
In Proofgold the corresponding term root is 7c0f0b... and proposition id is bb6d2d...
Axiom. (ordinal_In_Or_Subq) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 70fe0d... and proposition id is 7b99b0...
Axiom. (ordinal_linear) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 3f30a5... and proposition id is 3b1fce...
Axiom. (ordinal_ordsucc_In_eq) We take the following as an axiom:
∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta
In Proofgold the corresponding term root is 500378... and proposition id is 0c19e7...
Axiom. (ordinal_lim_or_succ) We take the following as an axiom:
∀alpha, ordinal alpha → (∀beta ∈ alpha, ordsucc beta ∈ alpha) ∨ (∃beta ∈ alpha, alpha = ordsucc beta)
In Proofgold the corresponding term root is 4450c0... and proposition id is a12697...
Axiom. (ordinal_ordsucc_In) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha
In Proofgold the corresponding term root is f4b1ed... and proposition id is 8ab1dc...
Axiom. (ordinal_famunion) We take the following as an axiom:
∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃x ∈ XF x)
In Proofgold the corresponding term root is fc0359... and proposition id is b3f1e1...
Axiom. (ordinal_binintersect) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)
In Proofgold the corresponding term root is 43d507... and proposition id is 025328...
Axiom. (ordinal_binunion) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)
In Proofgold the corresponding term root is 861ae3... and proposition id is b24989...
Axiom. (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
In Proofgold the corresponding term root is bb5575... and proposition id is 973cd8...
Axiom. (least_ordinal_ex) We take the following as an axiom:
∀p : set → prop, (∃alpha, ordinal alpha ∧ p alpha) → ∃alpha, ordinal alpha ∧ p alpha ∧ ∀beta ∈ alpha, ¬ p beta
In Proofgold the corresponding term root is df6c77... and proposition id is ab6fe1...
Definition. 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.
In Proofgold the corresponding term root is 264b04... and object id is 442958...
Definition. 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.
In Proofgold the corresponding term root is 76bef6... and object id is f7d93b...
Axiom. (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
In Proofgold the corresponding term root is a9064f... and proposition id is 3229f9...
Axiom. (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
In Proofgold the corresponding term root is 2cf278... and proposition id is cf0759...
Primitive. The name inv is a term of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 896fa9... and object id is f3a015...
Axiom. (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
In Proofgold the corresponding term root is ebf371... and proposition id is 6489d2...
Axiom. (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
In Proofgold the corresponding term root is 18ecce... and proposition id is d16557...
Axiom. (bij_inv) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
In Proofgold the corresponding term root is ff1332... and proposition id is 69c537...
Axiom. (bij_id) We take the following as an axiom:
∀X, bij X X (λx ⇒ x)
In Proofgold the corresponding term root is a8cefb... and proposition id is 707f37...
Axiom. (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))
In Proofgold the corresponding term root is abf851... and proposition id is 4c9045...
Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.
In Proofgold the corresponding term root is eb4419... and object id is 7f2bf6...
Axiom. (equip_ref) We take the following as an axiom:
∀X, equip X X
In Proofgold the corresponding term root is cc7e80... and proposition id is c41fb1...
Axiom. (equip_sym) We take the following as an axiom:
∀X Y, equip X Y → equip Y X
In Proofgold the corresponding term root is f28954... and proposition id is 272f04...
Axiom. (equip_tra) We take the following as an axiom:
∀X Y Z, equip X Y → equip Y Z → equip X Z
In Proofgold the corresponding term root is 322bfb... and proposition id is d616c1...
Axiom. (equip_0_Empty) We take the following as an axiom:
∀X, equip X 0 → X = 0
In Proofgold the corresponding term root is 83ab2f... and proposition id is 4c101c...
Beginning of Section SchroederBernstein
Axiom. (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
In Proofgold the corresponding term root is 2ffb9f... and proposition id is 31ef9d...
Axiom. (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
In Proofgold the corresponding term root is 6799e8... and proposition id is 43ed06...
Axiom. (image_monotone) We take the following as an axiom:
∀f : set → set, ∀U V, U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V}
In Proofgold the corresponding term root is ac6fe8... and proposition id is f923c6...
Axiom. (setminus_antimonotone) We take the following as an axiom:
∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U
In Proofgold the corresponding term root is 67bd0e... and proposition id is f9b735...
Axiom. (SchroederBernstein) We take the following as an axiom:
∀A B, ∀f g : set → set, inj A B f → inj B A g → equip A B
In Proofgold the corresponding term root is e247a8... and proposition id is 03a2cd...
End of Section SchroederBernstein
Beginning of Section PigeonHole
Axiom. (PigeonHole_nat) We take the following as an axiom:
∀n, nat_p n → ∀f : set → set, (∀i ∈ ordsucc n, f i ∈ n) → ¬ (∀i j ∈ ordsucc n, f i = f j → i = j)
In Proofgold the corresponding term root is d756b7... and proposition id is 604a84...
Axiom. (PigeonHole_nat_bij) We take the following as an axiom:
∀n, nat_p n → ∀f : set → set, (∀i ∈ n, f i ∈ n) → (∀i j ∈ n, f i = f j → i = j) → bij n n f
In Proofgold the corresponding term root is cd3b8d... and proposition id is 44a08f...
End of Section PigeonHole
Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.
In Proofgold the corresponding term root is 04632d... and object id is 4c54a5...
Axiom. (finite_ind) We take the following as an axiom:
∀p : set → prop, p Empty → (∀X y, finite X → y ∉ X → p X → p (X ∪ {y})) → ∀X, finite X → p X
In Proofgold the corresponding term root is c53cb1... and proposition id is 19baca...
Axiom. (finite_Empty) We take the following as an axiom:
finite 0
In Proofgold the corresponding term root is c073c6... and proposition id is ec2c27...
Axiom. (adjoin_finite) We take the following as an axiom:
∀X y, finite X → finite (X ∪ {y})
In Proofgold the corresponding term root is def317... and proposition id is e6aa6d...
Axiom. (binunion_finite) We take the following as an axiom:
∀X, finite X → ∀Y, finite Y → finite (X ∪ Y)
In Proofgold the corresponding term root is fc26bb... and proposition id is 3365a6...
Axiom. (famunion_nat_finite) We take the following as an axiom:
∀X : set → set, ∀n, nat_p n → (∀i ∈ n, finite (X i)) → finite (⋃i ∈ nX i)
In Proofgold the corresponding term root is 7f709a... and proposition id is f84ed0...
Axiom. (Subq_finite) We take the following as an axiom:
∀X, finite X → ∀Y, Y ⊆ X → finite Y
In Proofgold the corresponding term root is 6b062c... and proposition id is e4e3ee...
Axiom. (TransSet_In_ordsucc_Subq) We take the following as an axiom:
∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y
In Proofgold the corresponding term root is f80c6e... and proposition id is 341993...
Axiom. (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
In Proofgold the corresponding term root is 083f82... and proposition id is b1b26e...
Axiom. (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
In Proofgold the corresponding term root is c3e995... and proposition id is ed9d65...
Axiom. (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
In Proofgold the corresponding term root is 028a8f... and proposition id is 80b3b0...
Axiom. (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
In Proofgold the corresponding term root is 7807d4... and proposition id is d1536a...
Beginning of Section Descr_ii
Variable P : (set → set) → prop
Primitive. The name Descr_ii is a term of type set → set.
In Proofgold the corresponding term root is 3bae39... and object id is 3a4ad2...
Hypothesis Pex : ∃f : set → set, P f
Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
Axiom. (Descr_ii_prop) We take the following as an axiom:
P Descr_ii
In Proofgold the corresponding term root is 45bfc0... and proposition id is e705d1...
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set) → prop
Primitive. The name Descr_iii is a term of type set → set → set.
In Proofgold the corresponding term root is ca5fc1... and object id is 877e40...
Hypothesis Pex : ∃f : set → set → set, P f
Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
Axiom. (Descr_iii_prop) We take the following as an axiom:
P Descr_iii
In Proofgold the corresponding term root is 261c3d... and proposition id is 160421...
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1 → prop
Primitive. The name Descr_Vo1 is a term of type Vo 1.
In Proofgold the corresponding term root is 615c0a... and object id is a4ecdd...
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
Axiom. (Descr_Vo1_prop) We take the following as an axiom:
P Descr_Vo1
In Proofgold the corresponding term root is f550ef... and proposition id is 2213e2...
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Primitive. The name If_ii is a term of type set → set.
In Proofgold the corresponding term root is 17057f... and object id is e29015...
Axiom. (If_ii_1) We take the following as an axiom:
p → If_ii = f
In Proofgold the corresponding term root is 31521e... and proposition id is 342a1f...
Axiom. (If_ii_0) We take the following as an axiom:
¬ p → If_ii = g
In Proofgold the corresponding term root is 6acab0... and proposition id is 36418b...
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Primitive. The name If_iii is a term of type set → set → set.
In Proofgold the corresponding term root is 331476... and object id is dd7e2b...
Axiom. (If_iii_1) We take the following as an axiom:
p → If_iii = f
In Proofgold the corresponding term root is 2d62ee... and proposition id is 48bc10...
Axiom. (If_iii_0) We take the following as an axiom:
¬ p → If_iii = g
In Proofgold the corresponding term root is 9b77d9... and proposition id is d11201...
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set) → set
Definition. 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.
In Proofgold the corresponding term root is fde509... and object id is a81e18...
Primitive. The name In_rec_i is a term of type set → set.
In Proofgold the corresponding term root is fac413... and object id is 93aac4...
Axiom. (In_rec_i_G_c) We take the following as an axiom:
∀X : set, ∀f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) → In_rec_i_G X (F X f)
In Proofgold the corresponding term root is 4da597... and proposition id is bc452e...
Axiom. (In_rec_i_G_inv) We take the following as an axiom:
∀X : set, ∀Y : set, In_rec_i_G X Y → ∃f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is ba1774... and proposition id is 5cfac2...
Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_i_G_f) We take the following as an axiom:
∀X : set, ∀Y Z : set, In_rec_i_G X Y → In_rec_i_G X Z → Y = Z
In Proofgold the corresponding term root is 25f7ff... and proposition id is 02b4e4...
Axiom. (In_rec_i_G_In_rec_i) We take the following as an axiom:
∀X : set, In_rec_i_G X (In_rec_i X)
In Proofgold the corresponding term root is 095407... and proposition id is dc2359...
Axiom. (In_rec_i_G_In_rec_i_d) We take the following as an axiom:
∀X : set, In_rec_i_G X (F X In_rec_i)
In Proofgold the corresponding term root is b480fe... and proposition id is 5b92ff...
Axiom. (In_rec_i_eq) We take the following as an axiom:
∀X : set, In_rec_i X = F X In_rec_i
In Proofgold the corresponding term root is 110653... and proposition id is e8653b...
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set)) → (set → set)
Definition. 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.
In Proofgold the corresponding term root is 6edba9... and object id is b31dbf...
Primitive. The name In_rec_ii is a term of type set → (set → set).
In Proofgold the corresponding term root is f3c9ab... and object id is 3841bd...
Axiom. (In_rec_G_ii_c) We take the following as an axiom:
∀X : set, ∀f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) → In_rec_G_ii X (F X f)
In Proofgold the corresponding term root is 381613... and proposition id is 49fbeb...
Axiom. (In_rec_G_ii_inv) We take the following as an axiom:
∀X : set, ∀Y : (set → set), In_rec_G_ii X Y → ∃f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 336cfe... and proposition id is b4a180...
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_G_ii_f) We take the following as an axiom:
∀X : set, ∀Y Z : (set → set), In_rec_G_ii X Y → In_rec_G_ii X Z → Y = Z
In Proofgold the corresponding term root is 64b6e8... and proposition id is a3826b...
Axiom. (In_rec_G_ii_In_rec_ii) We take the following as an axiom:
∀X : set, In_rec_G_ii X (In_rec_ii X)
In Proofgold the corresponding term root is b2f81f... and proposition id is 2893f4...
Axiom. (In_rec_G_ii_In_rec_ii_d) We take the following as an axiom:
∀X : set, In_rec_G_ii X (F X In_rec_ii)
In Proofgold the corresponding term root is 2ff1ca... and proposition id is f12072...
Axiom. (In_rec_ii_eq) We take the following as an axiom:
∀X : set, In_rec_ii X = F X In_rec_ii
In Proofgold the corresponding term root is f7aff6... and proposition id is 7bb1e9...
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set → (set → (set → set → set)) → (set → set → set)
Definition. 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.
In Proofgold the corresponding term root is 533cc7... and object id is 7a2f6b...
Primitive. The name In_rec_iii is a term of type set → (set → set → set).
In Proofgold the corresponding term root is 9b3a85... and object id is 651077...
Axiom. (In_rec_G_iii_c) We take the following as an axiom:
∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) → In_rec_G_iii X (F X f)
In Proofgold the corresponding term root is 4ea572... and proposition id is a7d90f...
Axiom. (In_rec_G_iii_inv) We take the following as an axiom:
∀X : set, ∀Y : (set → set → set), In_rec_G_iii X Y → ∃f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 5f6e0c... and proposition id is f47dae...
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_G_iii_f) We take the following as an axiom:
∀X : set, ∀Y Z : (set → set → set), In_rec_G_iii X Y → In_rec_G_iii X Z → Y = Z
In Proofgold the corresponding term root is 574d71... and proposition id is 40978a...
Axiom. (In_rec_G_iii_In_rec_iii) We take the following as an axiom:
∀X : set, In_rec_G_iii X (In_rec_iii X)
In Proofgold the corresponding term root is d8c3ba... and proposition id is 37f9a7...
Axiom. (In_rec_G_iii_In_rec_iii_d) We take the following as an axiom:
∀X : set, In_rec_G_iii X (F X In_rec_iii)
In Proofgold the corresponding term root is 60c6d7... and proposition id is c05f77...
Axiom. (In_rec_iii_eq) We take the following as an axiom:
∀X : set, In_rec_iii X = F X In_rec_iii
In Proofgold the corresponding term root is 5f0574... and proposition id is a0709d...
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : set → set → set
Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
Definition. We define nat_primrec to be In_rec_i F of type set → set.
In Proofgold the corresponding term root is 3be1c5... and object id is 1a64d5...
Axiom. (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
In Proofgold the corresponding term root is c955cd... and proposition id is 9ba887...
Axiom. (nat_primrec_0) We take the following as an axiom:
nat_primrec 0 = z
In Proofgold the corresponding term root is 416eaf... and proposition id is be11a2...
Axiom. (nat_primrec_S) We take the following as an axiom:
∀n : set, nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n)
In Proofgold the corresponding term root is 797185... and proposition id is fe4c88...
End of Section NatRec
Beginning of Section NatArith
Definition. We define add_nat to be λn m : set ⇒ nat_primrec n (λ_ r ⇒ ordsucc r) m of type set → set → set.
In Proofgold the corresponding term root is afa8ae... and object id is 29768b...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Axiom. (add_nat_0R) We take the following as an axiom:
∀n : set, n + 0 = n
In Proofgold the corresponding term root is 402bcb... and proposition id is 23277f...
Axiom. (add_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)
In Proofgold the corresponding term root is a53523... and proposition id is e61e18...
Axiom. (add_nat_p) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n + m)
In Proofgold the corresponding term root is a68eaa... and proposition id is 4c1d5d...
Axiom. (add_nat_1_1_2) We take the following as an axiom:
1 + 1 = 2
In Proofgold the corresponding term root is 174b87... and proposition id is 77f23f...
Axiom. (add_nat_0L) We take the following as an axiom:
∀m : set, nat_p m → 0 + m = m
In Proofgold the corresponding term root is 8ceeb9... and proposition id is 84fbe4...
Axiom. (add_nat_SL) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n + m = ordsucc (n + m)
In Proofgold the corresponding term root is 9c442a... and proposition id is e6efb7...
Axiom. (add_nat_com) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → n + m = m + n
In Proofgold the corresponding term root is 40e952... and proposition id is 4ab179...
Axiom. (nat_Subq_add_ex) We take the following as an axiom:
∀n, nat_p n → ∀m, nat_p m → n ⊆ m → ∃k, nat_p k ∧ m = k + n
In Proofgold the corresponding term root is a1e406... and proposition id is 2c7fa4...
Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.
In Proofgold the corresponding term root is 35a1ef... and object id is 80ccc6...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Axiom. (mul_nat_0R) We take the following as an axiom:
∀n : set, n * 0 = 0
In Proofgold the corresponding term root is b2f56c... and proposition id is 1e5b2c...
Axiom. (mul_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n * ordsucc m = n + n * m
In Proofgold the corresponding term root is 37df54... and proposition id is 6dd6fc...
Axiom. (mul_nat_p) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n * m)
In Proofgold the corresponding term root is d63438... and proposition id is e910cb...
End of Section NatArith
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.
In Proofgold the corresponding term root is 8f0026... and object id is 7736ca...
Axiom. (Inj1_eq) We take the following as an axiom:
∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}
In Proofgold the corresponding term root is 73d189... and proposition id is b80457...
Axiom. (Inj1I1) We take the following as an axiom:
∀X : set, 0 ∈ Inj1 X
In Proofgold the corresponding term root is 3c2cc4... and proposition id is 4ccac3...
Axiom. (Inj1I2) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X
In Proofgold the corresponding term root is b6f23b... and proposition id is 086475...
Axiom. (Inj1E) We take the following as an axiom:
∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x
In Proofgold the corresponding term root is d4177b... and proposition id is 259217...
Axiom. (Inj1NE1) We take the following as an axiom:
∀x : set, Inj1 x ≠ 0
In Proofgold the corresponding term root is 5e3197... and proposition id is 09bd10...
Axiom. (Inj1NE2) We take the following as an axiom:
∀x : set, Inj1 x ∉ {0}
In Proofgold the corresponding term root is 9e94b1... and proposition id is 3c4dd0...
Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.
In Proofgold the corresponding term root is 814321... and object id is 7d54be...
Axiom. (Inj0I) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X
In Proofgold the corresponding term root is 7fc2e9... and proposition id is 565378...
Axiom. (Inj0E) We take the following as an axiom:
∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x
In Proofgold the corresponding term root is 416684... and proposition id is 855213...
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.
In Proofgold the corresponding term root is f202c1... and object id is 2e1675...
Axiom. (Unj_eq) We take the following as an axiom:
∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}
In Proofgold the corresponding term root is 3b1aa6... and proposition id is b4c852...
Axiom. (Unj_Inj1_eq) We take the following as an axiom:
∀X : set, Unj (Inj1 X) = X
In Proofgold the corresponding term root is e2cdfc... and proposition id is ae00e1...
Axiom. (Inj1_inj) We take the following as an axiom:
∀X Y : set, Inj1 X = Inj1 Y → X = Y
In Proofgold the corresponding term root is ce8b66... and proposition id is 3e918a...
Axiom. (Unj_Inj0_eq) We take the following as an axiom:
∀X : set, Unj (Inj0 X) = X
In Proofgold the corresponding term root is f3e39c... and proposition id is 3808ad...
Axiom. (Inj0_inj) We take the following as an axiom:
∀X Y : set, Inj0 X = Inj0 Y → X = Y
In Proofgold the corresponding term root is 662f53... and proposition id is 29f945...
Axiom. (Inj0_0) We take the following as an axiom:
Inj0 0 = 0
In Proofgold the corresponding term root is 171f47... and proposition id is 57f5af...
Axiom. (Inj0_Inj1_neq) We take the following as an axiom:
∀X Y : set, Inj0 X ≠ Inj1 Y
In Proofgold the corresponding term root is 21bb03... and proposition id is 2b14d8...
Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is afe37e... and object id is 5b7e09...
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
Axiom. (Inj0_setsum) We take the following as an axiom:
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
In Proofgold the corresponding term root is 1c6408... and proposition id is 246ac5...
Axiom. (Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
In Proofgold the corresponding term root is d47a0f... and proposition id is 2b4f94...
Axiom. (setsum_Inj_inv) We take the following as an axiom:
∀X Y z : set, z ∈ X + Y → (∃x ∈ X, z = Inj0 x) ∨ (∃y ∈ Y, z = Inj1 y)
In Proofgold the corresponding term root is 645a6c... and proposition id is a36400...
Axiom. (Inj0_setsum_0L) We take the following as an axiom:
∀X : set, 0 + X = Inj0 X
In Proofgold the corresponding term root is 0be1e7... and proposition id is 36fabd...
Axiom. (Subq_1_Sing0) We take the following as an axiom:
1 ⊆ {0}
In Proofgold the corresponding term root is aeae9a... and proposition id is abab2d...
Axiom. (Subq_Sing0_1) We take the following as an axiom:
{0} ⊆ 1
In Proofgold the corresponding term root is cd7864... and proposition id is 1806bf...
Axiom. (eq_1_Sing0) We take the following as an axiom:
1 = {0}
In Proofgold the corresponding term root is 5e5996... and proposition id is 264f38...
Axiom. (Inj1_setsum_1L) We take the following as an axiom:
∀X : set, 1 + X = Inj1 X
In Proofgold the corresponding term root is f8d721... and proposition id is 45b1c8...
Axiom. (nat_setsum1_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → 1 + n = ordsucc n
In Proofgold the corresponding term root is 3be3cc... and proposition id is 8fbdda...
Axiom. (setsum_0_0) We take the following as an axiom:
0 + 0 = 0
In Proofgold the corresponding term root is 5dcaeb... and proposition id is 4f3651...
Axiom. (setsum_1_0_1) We take the following as an axiom:
1 + 0 = 1
In Proofgold the corresponding term root is c2107f... and proposition id is e8609a...
Axiom. (setsum_1_1_2) We take the following as an axiom:
1 + 1 = 2
In Proofgold the corresponding term root is 9c89c0... and proposition id is ef3f41...
Beginning of Section pair_setsum
Let pair ≝ setsum
Definition. We define proj0 to be λZ ⇒ {Unj z|z ∈ Z, ∃x : set, Inj0 x = z} of type set → set.
In Proofgold the corresponding term root is 21a488... and object id is 9c5e4e...
Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.
In Proofgold the corresponding term root is 7aba21... and object id is 7fd81e...
Axiom. (Inj0_pair_0_eq) We take the following as an axiom:
Inj0 = pair 0
In Proofgold the corresponding term root is 0fe601... and proposition id is 490c2a...
Axiom. (Inj1_pair_1_eq) We take the following as an axiom:
Inj1 = pair 1
In Proofgold the corresponding term root is 0bbde8... and proposition id is 8272d7...
Axiom. (pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
In Proofgold the corresponding term root is 21a414... and proposition id is 34813b...
Axiom. (pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
In Proofgold the corresponding term root is bdaab5... and proposition id is b5248b...
Axiom. (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)
In Proofgold the corresponding term root is aac610... and proposition id is 0afdeb...
Axiom. (pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
In Proofgold the corresponding term root is 042f30... and proposition id is f50b25...
Axiom. (pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
In Proofgold the corresponding term root is 505cfc... and proposition id is 30a181...
Axiom. (proj0I) We take the following as an axiom:
∀w u : set, pair 0 u ∈ w → u ∈ proj0 w
In Proofgold the corresponding term root is be55b9... and proposition id is 840c8d...
Axiom. (proj0E) We take the following as an axiom:
∀w u : set, u ∈ proj0 w → pair 0 u ∈ w
In Proofgold the corresponding term root is b9afa9... and proposition id is 5b7f58...
Axiom. (proj1I) We take the following as an axiom:
∀w u : set, pair 1 u ∈ w → u ∈ proj1 w
In Proofgold the corresponding term root is fbddb7... and proposition id is 595b3c...
Axiom. (proj1E) We take the following as an axiom:
∀w u : set, u ∈ proj1 w → pair 1 u ∈ w
In Proofgold the corresponding term root is 06e32e... and proposition id is c3c3e6...
Axiom. (proj0_pair_eq) We take the following as an axiom:
∀X Y : set, proj0 (pair X Y) = X
In Proofgold the corresponding term root is d546f3... and proposition id is 0c12fb...
Axiom. (proj1_pair_eq) We take the following as an axiom:
∀X Y : set, proj1 (pair X Y) = Y
In Proofgold the corresponding term root is 9f98a8... and proposition id is e0cb0f...
Definition. We define Sigma to be λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 8acbb2... and object id is d64bcb...
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Axiom. (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
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Axiom. (Sigma_eta_proj0_proj1) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is afe962... and proposition id is 77ba8e...
Axiom. (proj_Sigma_eta) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z
In Proofgold the corresponding term root is 8b31cd... and proposition id is f30f76...
Axiom. (proj0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
In Proofgold the corresponding term root is ae12a7... and proposition id is 506869...
Axiom. (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)
In Proofgold the corresponding term root is 553db6... and proposition id is d79917...
Axiom. (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
In Proofgold the corresponding term root is c294a7... and proposition id is 81d8b0...
Axiom. (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
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.
In Proofgold the corresponding term root is fc0b60... and object id is 107b6b...
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Let lam : set → (set → set) → set ≝ Sigma
Definition. We define ap to be λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type set → set → set.
In Proofgold the corresponding term root is c7aaa6... and object id is a4d82c...
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.
Axiom. (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
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Axiom. (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
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Axiom. (apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
In Proofgold the corresponding term root is 64667c... and proposition id is 770816...
Axiom. (apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
In Proofgold the corresponding term root is 85693a... and proposition id is c08565...
Axiom. (beta) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
In Proofgold the corresponding term root is 95991d... and proposition id is b5182d...
Axiom. (proj0_ap_0) We take the following as an axiom:
∀u, proj0 u = u 0
In Proofgold the corresponding term root is 588f37... and proposition id is d4f351...
Axiom. (proj1_ap_1) We take the following as an axiom:
∀u, proj1 u = u 1
In Proofgold the corresponding term root is d116b3... and proposition id is 2328b0...
Axiom. (pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
In Proofgold the corresponding term root is ac34b9... and proposition id is bb0477...
Axiom. (pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
In Proofgold the corresponding term root is 2274fc... and proposition id is cce3d2...
Axiom. (ap0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
In Proofgold the corresponding term root is 861a79... and proposition id is 1affcc...
Axiom. (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))
In Proofgold the corresponding term root is ad3bfe... and proposition id is 520af5...
Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.
In Proofgold the corresponding term root is ade2bb... and object id is 5c8a35...
Axiom. (pair_p_I) We take the following as an axiom:
∀x y, pair_p (pair x y)
In Proofgold the corresponding term root is 06d9a2... and proposition id is 8f2fde...
Axiom. (Subq_2_UPair01) We take the following as an axiom:
2 ⊆ {0,1}
In Proofgold the corresponding term root is d90884... and proposition id is 94203e...
Axiom. (tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
In Proofgold the corresponding term root is c122f6... and proposition id is 18198e...
Definition. 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.
In Proofgold the corresponding term root is 40f1c3... and object id is a7b1b0...
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Axiom. (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
In Proofgold the corresponding term root is 2b7ef4... and proposition id is 1f8f6f...
Axiom. (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)
In Proofgold the corresponding term root is c0b88f... and proposition id is 3733f8...
Axiom. (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
In Proofgold the corresponding term root is 7c3617... and proposition id is cb962e...
Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.
In Proofgold the corresponding term root is 1de7fc... and object id is 021ae0...
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Axiom. (pair_tuple_fun) We take the following as an axiom:
pair = (λx y ⇒ (x,y))
In Proofgold the corresponding term root is b02dfa... and proposition id is ed37d0...
Axiom. (lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Beginning of Section Tuples
Variable x0 x1 : set
Axiom. (tuple_2_0_eq) We take the following as an axiom:
(x0,x1) 0 = x0
In Proofgold the corresponding term root is 42d452... and proposition id is ad9037...
Axiom. (tuple_2_1_eq) We take the following as an axiom:
(x0,x1) 1 = x1
In Proofgold the corresponding term root is 596569... and proposition id is 2c0ad1...
End of Section Tuples
Axiom. (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}
In Proofgold the corresponding term root is c486ab... and proposition id is f689a0...
Axiom. (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
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Axiom. (tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
In Proofgold the corresponding term root is e907b8... and proposition id is 1f9d7f...
End of Section pair_setsum