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Theorem cat1 18049
Description: The definition of category df-cat 17611 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18046 and setc2ohom 18047 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17978 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
Assertion
Ref Expression
cat1 𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ] ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤))
Distinct variable group:   𝑏,𝑐,,𝑤,𝑥,𝑦,𝑧

Proof of Theorem cat1
StepHypRef Expression
1 2on 8475 . . 3 2o ∈ On
2 eqid 2724 . . . 4 (SetCat‘2o) = (SetCat‘2o)
32setccat 18037 . . 3 (2o ∈ On → (SetCat‘2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCat‘2o) ∈ Cat
51a1i 11 . . . 4 (⊤ → 2o ∈ On)
6 eqid 2724 . . . 4 (Base‘(SetCat‘2o)) = (Base‘(SetCat‘2o))
7 eqid 2724 . . . 4 (Hom ‘(SetCat‘2o)) = (Hom ‘(SetCat‘2o))
8 0ex 5297 . . . . . . 7 ∅ ∈ V
98prid1 4758 . . . . . 6 ∅ ∈ {∅, {∅}}
10 df2o2 8470 . . . . . 6 2o = {∅, {∅}}
119, 10eleqtrri 2824 . . . . 5 ∅ ∈ 2o
1211a1i 11 . . . 4 (⊤ → ∅ ∈ 2o)
13 p0ex 5372 . . . . . . 7 {∅} ∈ V
1413prid2 4759 . . . . . 6 {∅} ∈ {∅, {∅}}
1514, 10eleqtrri 2824 . . . . 5 {∅} ∈ 2o
1615a1i 11 . . . 4 (⊤ → {∅} ∈ 2o)
17 0nep0 5346 . . . . 5 ∅ ≠ {∅}
1817a1i 11 . . . 4 (⊤ → ∅ ≠ {∅})
192, 5, 6, 7, 12, 16, 18cat1lem 18048 . . 3 (⊤ → ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
2019mptru 1540 . 2 𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))
21 fvexd 6896 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22 fveq2 6881 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23 fvexd 6896 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24 fveq2 6881 . . . . . 6 (𝑐 = (SetCat‘2o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
2524adantr 480 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
26 oveq 7407 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑥𝑦) = (𝑥(Hom ‘(SetCat‘2o))𝑦))
27 oveq 7407 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑧𝑤) = (𝑧(Hom ‘(SetCat‘2o))𝑤))
2826, 27ineq12d 4205 . . . . . . . . . . 11 ( = (Hom ‘(SetCat‘2o)) → ((𝑥𝑦) ∩ (𝑧𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
2928neeq1d 2992 . . . . . . . . . 10 ( = (Hom ‘(SetCat‘2o)) → (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅))
3029anbi1d 629 . . . . . . . . 9 ( = (Hom ‘(SetCat‘2o)) → ((((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
31302rexbidv 3211 . . . . . . . 8 ( = (Hom ‘(SetCat‘2o)) → (∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
32312rexbidv 3211 . . . . . . 7 ( = (Hom ‘(SetCat‘2o)) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
3332adantl 481 . . . . . 6 (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ = (Hom ‘(SetCat‘2o))) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
34 pm4.61 404 . . . . . . . . . . 11 (¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
35342rexbii 3121 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
36 rexnal2 3127 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
3735, 36bitr3i 277 . . . . . . . . 9 (∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
38372rexbii 3121 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
39 rexnal2 3127 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
4038, 39bitri 275 . . . . . . 7 (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
4140a1i 11 . . . . . 6 (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ = (Hom ‘(SetCat‘2o))) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤))))
42 rexeq 3313 . . . . . . . . . 10 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
43422rexbidv 3211 . . . . . . . . 9 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑦𝑏𝑧𝑏𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
4443rexbidv 3170 . . . . . . . 8 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
45 rexeq 3313 . . . . . . . . 9 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑧𝑏𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
46452rexbidv 3211 . . . . . . . 8 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
47 rexeq 3313 . . . . . . . . 9 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑦𝑏𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
4847rexeqbi1dv 3326 . . . . . . . 8 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥𝑏𝑦𝑏𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
4944, 46, 483bitrd 305 . . . . . . 7 (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
5049ad2antlr 724 . . . . . 6 (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ = (Hom ‘(SetCat‘2o))) → (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
5133, 41, 503bitr3d 309 . . . . 5 (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ = (Hom ‘(SetCat‘2o))) → (¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
5223, 25, 51sbcied2 3816 . . . 4 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → ([(Hom ‘𝑐) / ] ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
5321, 22, 52sbcied2 3816 . . 3 (𝑐 = (SetCat‘2o) → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ] ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
5453rspcev 3604 . 2 (((SetCat‘2o) ∈ Cat ∧ ∃𝑥 ∈ (Base‘(SetCat‘2o))∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ] ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
554, 20, 54mp2an 689 1 𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ] ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wtru 1534  wcel 2098  wne 2932  wral 3053  wrex 3062  Vcvv 3466  [wsbc 3769  cin 3939  c0 4314  {csn 4620  {cpr 4622  Oncon0 6354  cfv 6533  (class class class)co 7401  2oc2o 8455  Basecbs 17143  Hom chom 17207  Catccat 17607  SetCatcsetc 18027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-setc 18028
This theorem is referenced by: (None)
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