Theorem cat1 18055

Description: The definition of category df-cat 17625 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18052 and setc2ohom 18053 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17984 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 8408	. . 3 ⊢ 2o ∈ On
2		eqid 2739	. . . 4 ⊢ (SetCat‘2o) = (SetCat‘2o)
3	2	setccat 18043	. . 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 2739	. . . 4 ⊢ (Base‘(SetCat‘2o)) = (Base‘(SetCat‘2o))
7		eqid 2739	. . . 4 ⊢ (Hom ‘(SetCat‘2o)) = (Hom ‘(SetCat‘2o))
8		0ex 5229	. . . . . . 7 ⊢ ∅ ∈ V
9	8	prid1 4694	. . . . . 6 ⊢ ∅ ∈ {∅, {∅}}
10		df2o2 8404	. . . . . 6 ⊢ 2o = {∅, {∅}}
11	9, 10	eleqtrri 2838	. . . . 5 ⊢ ∅ ∈ 2o
12	11	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ 2o)
13		p0ex 5313	. . . . . . 7 ⊢ {∅} ∈ V
14	13	prid2 4695	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
15	14, 10	eleqtrri 2838	. . . . 5 ⊢ {∅} ∈ 2o
16	15	a1i 11	. . . 4 ⊢ (⊤ → {∅} ∈ 2o)
17		0nep0 5286	. . . . 5 ⊢ ∅ ≠ {∅}
18	17	a1i 11	. . . 4 ⊢ (⊤ → ∅ ≠ {∅})
19	2, 5, 6, 7, 12, 16, 18	cat1lem 18054	. . 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 1554	. 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 6842	. . . 4 ⊢ (𝑐 = (SetCat‘2o) → (Base‘𝑐) ∈ V)
22		fveq2 6827	. . . 4 ⊢ (𝑐 = (SetCat‘2o) → (Base‘𝑐) = (Base‘(SetCat‘2o)))
23		fvexd 6842	. . . . 5 ⊢ ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) ∈ V)
24		fveq2 6827	. . . . . 6 ⊢ (𝑐 = (SetCat‘2o) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
25	24	adantr 481	. . . . 5 ⊢ ((𝑐 = (SetCat‘2o) ∧ 𝑏 = (Base‘(SetCat‘2o))) → (Hom ‘𝑐) = (Hom ‘(SetCat‘2o)))
26		oveq 7362	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (𝑥ℎ𝑦) = (𝑥(Hom ‘(SetCat‘2o))𝑦))
27		oveq 7362	. . . . . . . . . . . 12 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (𝑧ℎ𝑤) = (𝑧(Hom ‘(SetCat‘2o))𝑤))
28	26, 27	ineq12d 4150	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → ((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) = ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)))
29	28	neeq1d 2993	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(SetCat‘2o)) → (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ↔ ((𝑥(Hom ‘(SetCat‘2o))𝑦) ∩ (𝑧(Hom ‘(SetCat‘2o))𝑤)) ≠ ∅))
30	29	anbi1d 637	. . . . . . . . 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 3204	. . . . . . . 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 3204	. . . . . . 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 482	. . . . . 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		pm4.61 405	. . . . . . . . . . 11 ⊢ (¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
35	34	2rexbii 3115	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 ¬ (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36		rexnal2 3121	. . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑏 ∃𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ¬ ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
38	37	2rexbii 3115	. . . . . . . 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 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)))
40	38, 39	bitri 276	. . . . . . 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 3293	. . . . . . . . . 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 3204	. . . . . . . . 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 3163	. . . . . . . 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 3293	. . . . . . . . 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 3204	. . . . . . . 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 3293	. . . . . . . . 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 3308	. . . . . . . 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 306	. . . . . . 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 733	. . . . . 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 310	. . . . 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 3767	. . . 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 3767	. . 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 3560	. 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 698	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 207   ∧ wa 396   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431  [wsbc 3723   ∩ cin 3882  ∅c0 4261  {csn 4555  {cpr 4557  Oncon0 6310  ‘cfv 6485  (class class class)co 7356  2_oc2o 8389  Basecbs 17170  Hom chom 17222  Catccat 17621  SetCatcsetc 18033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-setc 18034

This theorem is referenced by: (None)