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Theorem cat1 18142
Description: The definition of category df-cat 17711 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18139 and setc2ohom 18140 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18071 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 8520 . . 3 2o ∈ On
2 eqid 2737 . . . 4 (SetCat‘2o) = (SetCat‘2o)
32setccat 18130 . . 3 (2o ∈ On → (SetCat‘2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCat‘2o) ∈ Cat
51a1i 11 . . . 4 (⊤ → 2o ∈ On)
6 eqid 2737 . . . 4 (Base‘(SetCat‘2o)) = (Base‘(SetCat‘2o))
7 eqid 2737 . . . 4 (Hom ‘(SetCat‘2o)) = (Hom ‘(SetCat‘2o))
8 0ex 5307 . . . . . . 7 ∅ ∈ V
98prid1 4762 . . . . . 6 ∅ ∈ {∅, {∅}}
10 df2o2 8515 . . . . . 6 2o = {∅, {∅}}
119, 10eleqtrri 2840 . . . . 5 ∅ ∈ 2o
1211a1i 11 . . . 4 (⊤ → ∅ ∈ 2o)
13 p0ex 5384 . . . . . . 7 {∅} ∈ V
1413prid2 4763 . . . . . 6 {∅} ∈ {∅, {∅}}
1514, 10eleqtrri 2840 . . . . 5 {∅} ∈ 2o
1615a1i 11 . . . 4 (⊤ → {∅} ∈ 2o)
17 0nep0 5358 . . . . 5 ∅ ≠ {∅}
1817a1i 11 . . . 4 (⊤ → ∅ ≠ {∅})
192, 5, 6, 7, 12, 16, 18cat1lem 18141 . . 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 1547 . 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 6921 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22 fveq2 6906 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23 fvexd 6921 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24 fveq2 6906 . . . . . 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 7437 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑥𝑦) = (𝑥(Hom ‘(SetCat‘2o))𝑦))
27 oveq 7437 . . . . . . . . . . . 12 ( = (Hom ‘(SetCat‘2o)) → (𝑧𝑤) = (𝑧(Hom ‘(SetCat‘2o))𝑤))
2826, 27ineq12d 4221 . . . . . . . . . . 11 ( = (Hom ‘(SetCat‘2o)) → ((𝑥𝑦) ∩ (𝑧𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
2928neeq1d 3000 . . . . . . . . . 10 ( = (Hom ‘(SetCat‘2o)) → (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅))
3029anbi1d 631 . . . . . . . . 9 ( = (Hom ‘(SetCat‘2o)) → ((((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤))))
31302rexbidv 3222 . . . . . . . 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 3222 . . . . . . 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 3129 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
36 rexnal2 3135 . . . . . . . . . 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 3129 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
39 rexnal2 3135 . . . . . . . 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 3322 . . . . . . . . . 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 3222 . . . . . . . . 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 3179 . . . . . . . 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 3322 . . . . . . . . 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 3222 . . . . . . . 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 3322 . . . . . . . . 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 3339 . . . . . . . 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 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 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 3833 . . . 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 3833 . . 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 3622 . 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 206  wa 395   = wceq 1540  wtru 1541  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  [wsbc 3788  cin 3950  c0 4333  {csn 4626  {cpr 4628  Oncon0 6384  cfv 6561  (class class class)co 7431  2oc2o 8500  Basecbs 17247  Hom chom 17308  Catccat 17707  SetCatcsetc 18120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-setc 18121
This theorem is referenced by: (None)
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