Theorem cat1 18110

Description: The definition of category df-cat 17680 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18107 and setc2ohom 18108 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18039 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 ⊢ 2o ∈ On
2		eqid 2735	. . . 4 ⊢ (SetCat‘2o) = (SetCat‘2o)
3	2	setccat 18098	. . 3 ⊢ (2o ∈ On → (SetCat‘2o) ∈ Cat)
4	1, 3	ax-mp 5	. 2 ⊢ (SetCat‘2o) ∈ Cat
5	1	a1i 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 5277	. . . . . . 7 ⊢ ∅ ∈ V
9	8	prid1 4738	. . . . . 6 ⊢ ∅ ∈ {∅, {∅}}
10		df2o2 8489	. . . . . 6 ⊢ 2o = {∅, {∅}}
11	9, 10	eleqtrri 2833	. . . . 5 ⊢ ∅ ∈ 2o
12	11	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ 2o)
13		p0ex 5354	. . . . . . 7 ⊢ {∅} ∈ V
14	13	prid2 4739	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
15	14, 10	eleqtrri 2833	. . . . 5 ⊢ {∅} ∈ 2o
16	15	a1i 11	. . . 4 ⊢ (⊤ → {∅} ∈ 2o)
17		0nep0 5328	. . . . 5 ⊢ ∅ ≠ {∅}
18	17	a1i 11	. . . 4 ⊢ (⊤ → ∅ ≠ {∅})
19	2, 5, 6, 7, 12, 16, 18	cat1lem 18109	. . 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 = 𝑧 ∧ 𝑦 = 𝑤)))
20	19	mptru 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 = 𝑧 ∧ 𝑦 = 𝑤))
21		fvexd 6891	. . . 4 ⊢ (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22		fveq2 6876	. . . 4 ⊢ (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23		fvexd 6891	. . . . 5 ⊢ ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24		fveq2 6876	. . . . . 6 ⊢ (𝑐 = (SetCat‘2o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
25	24	adantr 480	. . . . 5 ⊢ ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
26		oveq 7411	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (𝑥ℎ𝑦) = (𝑥(Hom ‘(SetCat‘2o))𝑦))
27		oveq 7411	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (𝑧ℎ𝑤) = (𝑧(Hom ‘(SetCat‘2o))𝑤))
28	26, 27	ineq12d 4196	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → ((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
29	28	neeq1d 2991	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅))
30	29	anbi1d 631	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → ((((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
31	30	2rexbidv 3206	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
32	31	2rexbidv 3206	. . . . . . 7 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (∃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‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
33	32	adantl 481	. . . . . 6 ⊢ (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ ℎ = (Hom ‘(SetCat‘2o))) → (∃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‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))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 3116	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36		rexnal2 3122	. . . . . . . . . 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 3116	. . . . . . . 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 3122	. . . . . . . 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‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ ℎ = (Hom ‘(SetCat‘2o))) → (∃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 3301	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
43	42	2rexbidv 3206	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
44	43	rexbidv 3164	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
45		rexeq 3301	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
46	45	2rexbidv 3206	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
47		rexeq 3301	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑦 ∈ (Base‘(SetCat‘2o))∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
48	47	rexeqbi1dv 3318	. . . . . . . 8 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ (Base‘(SetCat‘2o))∃𝑤 ∈ (Base‘(SetCat‘2o))(((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
49	44, 46, 48	3bitrd 305	. . . . . . 7 ⊢ (𝑏 = (Base‘(SetCat‘2o)) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
50	49	ad2antlr 727	. . . . . 6 ⊢ (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ ℎ = (Hom ‘(SetCat‘2o))) → (∃𝑥 ∈ 𝑏 ∃𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
51	33, 41, 50	3bitr3d 309	. . . . 5 ⊢ (((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) ∧ ℎ = (Hom ‘(SetCat‘2o))) → (¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
52	23, 25, 51	sbcied2 3810	. . . 4 ⊢ ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → ([(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
53	21, 22, 52	sbcied2 3810	. . 3 ⊢ (𝑐 = (SetCat‘2o) → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 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 = 𝑧 ∧ 𝑦 = 𝑤))))
54	53	rspcev 3601	. 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 = 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 692	1 ⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3459  [wsbc 3765   ∩ cin 3925  ∅c0 4308  {csn 4601  {cpr 4603  Oncon0 6352  ‘cfv 6531  (class class class)co 7405  2_oc2o 8474  Basecbs 17228  Hom chom 17282  Catccat 17676  SetCatcsetc 18088

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17295  df-cco 17296  df-cat 17680  df-cid 17681  df-setc 18089

This theorem is referenced by: (None)