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Theorem cat1 18046

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 18043 and setc2ohom 18044 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17975 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 8479	. . 3 ⊢ 2_o ∈ On
2		eqid 2732	. . . 4 ⊢ (SetCat‘2_o) = (SetCat‘2_o)
3	2	setccat 18034	. . 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 2732	. . . 4 ⊢ (Base‘(SetCat‘2_o)) = (Base‘(SetCat‘2_o))
7		eqid 2732	. . . 4 ⊢ (Hom ‘(SetCat‘2_o)) = (Hom ‘(SetCat‘2_o))
8		0ex 5307	. . . . . . 7 ⊢ ∅ ∈ V
9	8	prid1 4766	. . . . . 6 ⊢ ∅ ∈ {∅, {∅}}
10		df2o2 8474	. . . . . 6 ⊢ 2_o = {∅, {∅}}
11	9, 10	eleqtrri 2832	. . . . 5 ⊢ ∅ ∈ 2_o
12	11	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ 2_o)
13		p0ex 5382	. . . . . . 7 ⊢ {∅} ∈ V
14	13	prid2 4767	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
15	14, 10	eleqtrri 2832	. . . . 5 ⊢ {∅} ∈ 2_o
16	15	a1i 11	. . . 4 ⊢ (⊤ → {∅} ∈ 2_o)
17		0nep0 5356	. . . . 5 ⊢ ∅ ≠ {∅}
18	17	a1i 11	. . . 4 ⊢ (⊤ → ∅ ≠ {∅})
19	2, 5, 6, 7, 12, 16, 18	cat1lem 18045	. . 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 1548	. 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 6906	. . . 4 ⊢ (𝑐 = (SetCat‘2_o) → (Base‘𝑐) ∈ V)
22		fveq2 6891	. . . 4 ⊢ (𝑐 = (SetCat‘2_o) → (Base‘𝑐) = (Base‘(SetCat‘2_o)))
23		fvexd 6906	. . . . 5 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → (Hom ‘𝑐) ∈ V)
24		fveq2 6891	. . . . . 6 ⊢ (𝑐 = (SetCat‘2_o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2_o)))
25	24	adantr 481	. . . . 5 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2_o)))
26		oveq 7414	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑥ℎ𝑦) = (𝑥(Hom ‘(SetCat‘2_o))𝑦))
27		oveq 7414	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑧ℎ𝑤) = (𝑧(Hom ‘(SetCat‘2_o))𝑤))
28	26, 27	ineq12d 4213	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → ((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) = ((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)))
29	28	neeq1d 3000	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)) ≠ ∅))
30	29	anbi1d 630	. . . . . . . . 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 3219	. . . . . . . 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 3219	. . . . . . 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 482	. . . . . 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		pm4.61 405	. . . . . . . . . . 11 ⊢ (¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
35	34	2rexbii 3129	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36		rexnal2 3135	. . . . . . . . . 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 276	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
38	37	2rexbii 3129	. . . . . . . 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 3135	. . . . . . . 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 274	. . . . . . 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 3321	. . . . . . . . . 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 3219	. . . . . . . . 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 3178	. . . . . . . 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 3321	. . . . . . . . 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 3219	. . . . . . . 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 3321	. . . . . . . . 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 3334	. . . . . . . 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 304	. . . . . . 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 725	. . . . . 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 308	. . . . 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 3824	. . . 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 3824	. . 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 3612	. 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 690	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 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 [wsbc 3777 ∩ cin 3947 ∅c0 4322 {csn 4628 {cpr 4630 Oncon0 6364 ‘cfv 6543 (class class class)co 7408 2_oc2o 8459 Basecbs 17143 Hom chom 17207 Catccat 17607 SetCatcsetc 18024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 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 18025

This theorem is referenced by: (None)

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