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Theorem cat1 18079

Description: The definition of category df-cat 17641 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18076 and setc2ohom 18077 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18008 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 8494	. . 3 ⊢ 2_o ∈ On
2		eqid 2728	. . . 4 ⊢ (SetCat‘2_o) = (SetCat‘2_o)
3	2	setccat 18067	. . 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 2728	. . . 4 ⊢ (Base‘(SetCat‘2_o)) = (Base‘(SetCat‘2_o))
7		eqid 2728	. . . 4 ⊢ (Hom ‘(SetCat‘2_o)) = (Hom ‘(SetCat‘2_o))
8		0ex 5301	. . . . . . 7 ⊢ ∅ ∈ V
9	8	prid1 4762	. . . . . 6 ⊢ ∅ ∈ {∅, {∅}}
10		df2o2 8489	. . . . . 6 ⊢ 2_o = {∅, {∅}}
11	9, 10	eleqtrri 2828	. . . . 5 ⊢ ∅ ∈ 2_o
12	11	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ 2_o)
13		p0ex 5378	. . . . . . 7 ⊢ {∅} ∈ V
14	13	prid2 4763	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
15	14, 10	eleqtrri 2828	. . . . 5 ⊢ {∅} ∈ 2_o
16	15	a1i 11	. . . 4 ⊢ (⊤ → {∅} ∈ 2_o)
17		0nep0 5352	. . . . 5 ⊢ ∅ ≠ {∅}
18	17	a1i 11	. . . 4 ⊢ (⊤ → ∅ ≠ {∅})
19	2, 5, 6, 7, 12, 16, 18	cat1lem 18078	. . 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 1541	. 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 480	. . . . 5 ⊢ ((𝑐 = (SetCat‘2_o) ∧ 𝑏 = (Base‘(SetCat‘2_o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2_o)))
26		oveq 7420	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑥ℎ𝑦) = (𝑥(Hom ‘(SetCat‘2_o))𝑦))
27		oveq 7420	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → (𝑧ℎ𝑤) = (𝑧(Hom ‘(SetCat‘2_o))𝑤))
28	26, 27	ineq12d 4209	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(SetCat‘2_o)) → ((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) = ((𝑥(Hom ‘(SetCat‘2_o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2_o))𝑤)))
29	28	neeq1d 2996	. . . . . . . . . 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 3215	. . . . . . . 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 3215	. . . . . . 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 3125	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36		rexnal2 3131	. . . . . . . . . 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 3125	. . . . . . . 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 3131	. . . . . . . 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 3317	. . . . . . . . . 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 3215	. . . . . . . . 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 3174	. . . . . . . 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 3317	. . . . . . . . 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 3215	. . . . . . . 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 3317	. . . . . . . . 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 3330	. . . . . . . 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 726	. . . . . 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 3822	. . . 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 3822	. . 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 3608	. 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 691	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 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 Vcvv 3470 [wsbc 3775 ∩ cin 3944 ∅c0 4318 {csn 4624 {cpr 4626 Oncon0 6363 ‘cfv 6542 (class class class)co 7414 2_oc2o 8474 Basecbs 17173 Hom chom 17237 Catccat 17637 SetCatcsetc 18057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-hom 17250 df-cco 17251 df-cat 17641 df-cid 17642 df-setc 18058

This theorem is referenced by: (None)

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