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Theorem cat1 18051

Description: The definition of category df-cat 17613 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18048 and setc2ohom 18049 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17980 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)

**Assertion**
Ref	Expression
cat1	⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))

Distinct variable group: 𝑏,𝑐,ℎ,𝑤,𝑥,𝑦,𝑧

**Proof of Theorem cat1**
Step	Hyp	Ref	Expression
1		2on 8476	. . 3 ⊢ 2_o ∈ On
2		eqid 2724	. . . 4 ⊢ (SetCat‘2_o) = (SetCat‘2_o)
3	2	setccat 18039	. . 3 ⊢ (2_o ∈ On → (SetCat‘2_o) ∈ Cat)
4	1, 3	ax-mp 5	. 2 ⊢ (SetCat‘2_o) ∈ Cat
5	1	a1i 11	. . . 4 ⊢ (⊤ → 2_o ∈ On)
6		eqid 2724	. . . 4 ⊢ (Base‘(SetCat‘2_o)) = (Base‘(SetCat‘2_o))
7		eqid 2724	. . . 4 ⊢ (Hom ‘(SetCat‘2_o)) = (Hom ‘(SetCat‘2_o))
8		0ex 5298	. . . . . . 7 ⊢ ∅ ∈ V
9	8	prid1 4759	. . . . . 6 ⊢ ∅ ∈ {∅, {∅}}
10		df2o2 8471	. . . . . 6 ⊢ 2_o = {∅, {∅}}
11	9, 10	eleqtrri 2824	. . . . 5 ⊢ ∅ ∈ 2_o
12	11	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ 2_o)
13		p0ex 5373	. . . . . . 7 ⊢ {∅} ∈ V
14	13	prid2 4760	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
15	14, 10	eleqtrri 2824	. . . . 5 ⊢ {∅} ∈ 2_o
16	15	a1i 11	. . . 4 ⊢ (⊤ → {∅} ∈ 2_o)
17		0nep0 5347	. . . . 5 ⊢ ∅ ≠ {∅}
18	17	a1i 11	. . . 4 ⊢ (⊤ → ∅ ≠ {∅})
19	2, 5, 6, 7, 12, 16, 18	cat1lem 18050	. . 3 ⊢ (⊤ → ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
20	19	mptru 1540	. 2 ⊢ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
21		fvexd 6897	. . . 4 ⊢ (𝑐 = (SetCat‘2_o) → (Base‘𝑐) ∈ V)
22		fveq2 6882	. . . 4 ⊢ (𝑐 = (SetCat‘2_o) → (Base‘𝑐) = (Base‘(SetCat‘2_o)))
23		fvexd 6897	. . . . 5 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → (Hom ‘𝑐) ∈ V)
24		fveq2 6882	. . . . . 6 ⊢ (𝑐 = (SetCat‘2_o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2_o)))
25	24	adantr 480	. . . . 5 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2_o)))
26		oveq 7408	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑥ℎ𝑦) = (𝑥(Hom ‘(SetCat‘2_o))𝑦))
27		oveq 7408	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑧ℎ𝑤) = (𝑧(Hom ‘(SetCat‘2_o))𝑤))
28	26, 27	ineq12d 4206	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → ((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) = ((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)))
29	28	neeq1d 2992	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅))
30	29	anbi1d 629	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → ((((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
31	30	2rexbidv 3211	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
32	31	2rexbidv 3211	. . . . . . 7 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
33	32	adantl 481	. . . . . 6 ⊢ (((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) ∧ ℎ = (Hom ‘(SetCat‘2_o))) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
34		pm4.61 404	. . . . . . . . . . 11 ⊢ (¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
35	34	2rexbii 3121	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36		rexnal2 3127	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
37	35, 36	bitr3i 277	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
38	37	2rexbii 3121	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
39		rexnal2 3127	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
40	38, 39	bitri 275	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
41	40	a1i 11	. . . . . 6 ⊢ (((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) ∧ ℎ = (Hom ‘(SetCat‘2_o))) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
42		rexeq 3313	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
43	42	2rexbidv 3211	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
44	43	rexbidv 3170	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
45		rexeq 3313	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
46	45	2rexbidv 3211	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
47		rexeq 3313	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
48	47	rexeqbi1dv 3326	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
49	44, 46, 48	3bitrd 305	. . . . . . 7 ⊢ (𝑏 = (Base‘(SetCat‘2_o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
50	49	ad2antlr 724	. . . . . 6 ⊢ (((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) ∧ ℎ = (Hom ‘(SetCat‘2_o))) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
51	33, 41, 50	3bitr3d 309	. . . . 5 ⊢ (((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) ∧ ℎ = (Hom ‘(SetCat‘2_o))) → (¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
52	23, 25, 51	sbcied2 3817	. . . 4 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → ([(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
53	21, 22, 52	sbcied2 3817	. . 3 ⊢ (𝑐 = (SetCat‘2_o) → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
54	53	rspcev 3604	. 2 ⊢ (((SetCat‘2_o) ∈ Cat ∧ ∃𝑥 ∈ (Base‘(SetCat‘2_o))∃𝑦 ∈ (Base‘(SetCat‘2_o))∃𝑧 ∈ (Base‘(SetCat‘2_o))∃𝑤 ∈ (Base‘(SetCat‘2_o))(((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
55	4, 20, 54	mp2an 689	1 ⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 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 3770 ∩ cin 3940 ∅c0 4315 {csn 4621 {cpr 4623 Oncon0 6355 ‘cfv 6534 (class class class)co 7402 2_oc2o 8456 Basecbs 17145 Hom chom 17209 Catccat 17609 SetCatcsetc 18029

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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-hom 17222 df-cco 17223 df-cat 17613 df-cid 17614 df-setc 18030

This theorem is referenced by: (None)

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