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Theorem cat1 17481
Description: The definition of category df-cat 17054 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 17478 and setc2ohom 17479 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17410 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 8151 . . 3 2o ∈ On
2 eqid 2739 . . . 4 (SetCat‘2o) = (SetCat‘2o)
32setccat 17469 . . 3 (2o ∈ On → (SetCat‘2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCat‘2o) ∈ Cat
51a1i 11 . . . 4 (⊤ → 2o ∈ On)
6 eqid 2739 . . . 4 (Base‘(SetCat‘2o)) = (Base‘(SetCat‘2o))
7 eqid 2739 . . . 4 (Hom ‘(SetCat‘2o)) = (Hom ‘(SetCat‘2o))
8 0ex 5185 . . . . . . 7 ∅ ∈ V
98prid1 4663 . . . . . 6 ∅ ∈ {∅, {∅}}
10 df2o2 8159 . . . . . 6 2o = {∅, {∅}}
119, 10eleqtrri 2833 . . . . 5 ∅ ∈ 2o
1211a1i 11 . . . 4 (⊤ → ∅ ∈ 2o)
13 p0ex 5261 . . . . . . 7 {∅} ∈ V
1413prid2 4664 . . . . . 6 {∅} ∈ {∅, {∅}}
1514, 10eleqtrri 2833 . . . . 5 {∅} ∈ 2o
1615a1i 11 . . . 4 (⊤ → {∅} ∈ 2o)
17 0nep0 5234 . . . . 5 ∅ ≠ {∅}
1817a1i 11 . . . 4 (⊤ → ∅ ≠ {∅})
192, 5, 6, 7, 12, 16, 18cat1lem 17480 . . 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 1549 . 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 6701 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22 fveq2 6686 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23 fvexd 6701 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24 fveq2 6686 . . . . . 6 (𝑐 = (SetCat‘2o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
2524adantr 484 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
26 oveq 7188 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑥𝑦) = (𝑥(Hom ‘(SetCat‘2o))𝑦))
27 oveq 7188 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑧𝑤) = (𝑧(Hom ‘(SetCat‘2o))𝑤))
2826, 27ineq12d 4114 . . . . . . . . . . 11 ( = (Hom ‘(SetCat‘2o)) → ((𝑥𝑦) ∩ (𝑧𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
2928neeq1d 2994 . . . . . . . . . 10 ( = (Hom ‘(SetCat‘2o)) → (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅))
3029anbi1d 633 . . . . . . . . 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 485 . . . . . 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 pm4.61 408 . . . . . . . . . . 11 (¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
35342rexbii 3163 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
36 rexnal2 3171 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
3735, 36bitr3i 280 . . . . . . . . 9 (∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
38372rexbii 3163 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
39 rexnal2 3171 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ¬ ∀𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
4038, 39bitri 278 . . . . . . 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 3312 . . . . . . . . . 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 3208 . . . . . . . 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 3312 . . . . . . . . 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 3312 . . . . . . . . 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 3310 . . . . . . . 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 308 . . . . . . 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 727 . . . . . 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 312 . . . . 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 3732 . . . 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 3732 . . 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 3529 . 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 692 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 209  wa 399   = wceq 1542  wtru 1543  wcel 2114  wne 2935  wral 3054  wrex 3055  Vcvv 3400  [wsbc 3685  cin 3852  c0 4221  {csn 4526  {cpr 4528  Oncon0 6182  cfv 6349  (class class class)co 7182  2oc2o 8137  Basecbs 16598  Hom chom 16691  Catccat 17050  SetCatcsetc 17459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-2o 8144  df-er 8332  df-map 8451  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-3 11792  df-4 11793  df-5 11794  df-6 11795  df-7 11796  df-8 11797  df-9 11798  df-n0 11989  df-z 12075  df-dec 12192  df-uz 12337  df-fz 12994  df-struct 16600  df-ndx 16601  df-slot 16602  df-base 16604  df-hom 16704  df-cco 16705  df-cat 17054  df-cid 17055  df-setc 17460
This theorem is referenced by: (None)
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