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Theorem cat1 18151
Description: The definition of category df-cat 17713 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18148 and setc2ohom 18149 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18080 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 8519 . . 3 2o ∈ On
2 eqid 2735 . . . 4 (SetCat‘2o) = (SetCat‘2o)
32setccat 18139 . . 3 (2o ∈ On → (SetCat‘2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCat‘2o) ∈ Cat
51a1i 11 . . . 4 (⊤ → 2o ∈ On)
6 eqid 2735 . . . 4 (Base‘(SetCat‘2o)) = (Base‘(SetCat‘2o))
7 eqid 2735 . . . 4 (Hom ‘(SetCat‘2o)) = (Hom ‘(SetCat‘2o))
8 0ex 5313 . . . . . . 7 ∅ ∈ V
98prid1 4767 . . . . . 6 ∅ ∈ {∅, {∅}}
10 df2o2 8514 . . . . . 6 2o = {∅, {∅}}
119, 10eleqtrri 2838 . . . . 5 ∅ ∈ 2o
1211a1i 11 . . . 4 (⊤ → ∅ ∈ 2o)
13 p0ex 5390 . . . . . . 7 {∅} ∈ V
1413prid2 4768 . . . . . 6 {∅} ∈ {∅, {∅}}
1514, 10eleqtrri 2838 . . . . 5 {∅} ∈ 2o
1615a1i 11 . . . 4 (⊤ → {∅} ∈ 2o)
17 0nep0 5364 . . . . 5 ∅ ≠ {∅}
1817a1i 11 . . . 4 (⊤ → ∅ ≠ {∅})
192, 5, 6, 7, 12, 16, 18cat1lem 18150 . . 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 1544 . 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 6922 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22 fveq2 6907 . . . 4 (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23 fvexd 6922 . . . . 5 ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24 fveq2 6907 . . . . . 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 4229 . . . . . . . . . . 11 ( = (Hom ‘(SetCat‘2o)) → ((𝑥𝑦) ∩ (𝑧𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
2928neeq1d 2998 . . . . . . . . . 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 3220 . . . . . . . 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 3220 . . . . . . 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 3127 . . . . . . . . . 10 (∃𝑧𝑏𝑤𝑏 ¬ (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)))
36 rexnal2 3133 . . . . . . . . . 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 3127 . . . . . . . 8 (∃𝑥𝑏𝑦𝑏𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑥𝑏𝑦𝑏 ¬ ∀𝑧𝑏𝑤𝑏 (((𝑥𝑦) ∩ (𝑧𝑤)) ≠ ∅ → (𝑥 = 𝑧𝑦 = 𝑤)))
39 rexnal2 3133 . . . . . . . 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 3320 . . . . . . . . . 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 3220 . . . . . . . . 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 3177 . . . . . . . 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 3320 . . . . . . . . 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 3220 . . . . . . . 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 3320 . . . . . . . . 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 3337 . . . . . . . 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 3839 . . . 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 3839 . . 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 1537  wtru 1538  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  cin 3962  c0 4339  {csn 4631  {cpr 4633  Oncon0 6386  cfv 6563  (class class class)co 7431  2oc2o 8499  Basecbs 17245  Hom chom 17309  Catccat 17709  SetCatcsetc 18129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-setc 18130
This theorem is referenced by: (None)
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