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Theorem hausdiag 23674

Description: A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

**Hypothesis**
Ref	Expression
hausdiag.x	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
hausdiag	⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽))))

Proof of Theorem hausdiag Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hausdiag.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	ishaus 23351	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))))
3		txtop 23598	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×_t 𝐽) ∈ Top)
4	3	anidms 566	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ×_t 𝐽) ∈ Top)
5		idssxp 6078	. . . . . 6 ⊢ ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋)
6	1, 1	txuni 23621	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
7	6	anidms 566	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
8	5, 7	sseqtrid 4061	. . . . 5 ⊢ (𝐽 ∈ Top → ( I ↾ 𝑋) ⊆ ∪ (𝐽 ×_t 𝐽))
9		eqid 2740	. . . . . 6 ⊢ ∪ (𝐽 ×_t 𝐽) = ∪ (𝐽 ×_t 𝐽)
10	9	iscld2 23057	. . . . 5 ⊢ (((𝐽 ×_t 𝐽) ∈ Top ∧ ( I ↾ 𝑋) ⊆ ∪ (𝐽 ×_t 𝐽)) → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽)) ↔ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×_t 𝐽)))
11	4, 8, 10	syl2anc 583	. . . 4 ⊢ (𝐽 ∈ Top → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽)) ↔ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×_t 𝐽)))
12		eltx 23597	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ((∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×_t 𝐽) ↔ ∀𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))
13	12	anidms 566	. . . 4 ⊢ (𝐽 ∈ Top → ((∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×_t 𝐽) ↔ ∀𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))
14		eldif 3986	. . . . . . . . . 10 ⊢ (𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ ∪ (𝐽 ×_t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))
15	7	eqcomd 2746	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ∪ (𝐽 ×_t 𝐽) = (𝑋 × 𝑋))
16	15	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → (𝑒 ∈ ∪ (𝐽 ×_t 𝐽) ↔ 𝑒 ∈ (𝑋 × 𝑋)))
17	16	anbi1d 630	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ((𝑒 ∈ ∪ (𝐽 ×_t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
18	14, 17	bitrid 283	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → (𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
19	18	imbi1d 341	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))))
20		impexp 450	. . . . . . . 8 ⊢ (((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))))
21	19, 20	bitrdi 287	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))))
22	21	ralbidv2 3180	. . . . . 6 ⊢ (𝐽 ∈ Top → (∀𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))))
23		eleq1 2832	. . . . . . . . 9 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ ( I ↾ 𝑋) ↔ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
24	23	notbid 318	. . . . . . . 8 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
25		eleq1 2832	. . . . . . . . . 10 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ (𝑐 × 𝑑) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑)))
26	25	anbi1d 630	. . . . . . . . 9 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → ((𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))
27	26	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → (∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))
28	24, 27	imbi12d 344	. . . . . . 7 ⊢ (𝑒 = ⟨𝑎, 𝑏⟩ → ((¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))))
29	28	ralxp 5866	. . . . . 6 ⊢ (∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))))
30	22, 29	bitrdi 287	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))))
31		vex 3492	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
32	31	opelresi 6017	. . . . . . . . . 10 ⊢ (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ))
33		ibar 528	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑋 → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎 ∈ 𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
34	33	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎 ∈ 𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
35		df-br 5167	. . . . . . . . . . . 12 ⊢ (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
36	31	ideq 5877	. . . . . . . . . . . 12 ⊢ (𝑎 I 𝑏 ↔ 𝑎 = 𝑏)
37	35, 36	bitr3i 277	. . . . . . . . . . 11 ⊢ (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
38	34, 37	bitr3di 286	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝑎 ∈ 𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ) ↔ 𝑎 = 𝑏))
39	32, 38	bitrid 283	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
40	39	adantl 481	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
41	40	necon3bbid 2984	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 ≠ 𝑏))
42		elssuni 4961	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐽 → 𝑐 ⊆ ∪ 𝐽)
43		elssuni 4961	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ 𝐽 → 𝑑 ⊆ ∪ 𝐽)
44		xpss12 5715	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ⊆ ∪ 𝐽 ∧ 𝑑 ⊆ ∪ 𝐽) → (𝑐 × 𝑑) ⊆ (∪ 𝐽 × ∪ 𝐽))
45	42, 43, 44	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽) → (𝑐 × 𝑑) ⊆ (∪ 𝐽 × ∪ 𝐽))
46	1, 1	xpeq12i 5728	. . . . . . . . . . . . . . 15 ⊢ (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽)
47	45, 46	sseqtrrdi 4060	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
48	47	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
49	7	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
50	48, 49	sseqtrd 4049	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑐 × 𝑑) ⊆ ∪ (𝐽 ×_t 𝐽))
51		reldisj 4476	. . . . . . . . . . . 12 ⊢ ((𝑐 × 𝑑) ⊆ ∪ (𝐽 ×_t 𝐽) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))
53		df-res 5712	. . . . . . . . . . . . . . 15 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
54	53	ineq2i 4238	. . . . . . . . . . . . . 14 ⊢ ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
55		inass 4249	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
56		inss1 4258	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑)
57	56, 48	sstrid 4020	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋))
58		ssv 4033	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 ⊆ V
59		xpss2 5720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V))
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 × 𝑋) ⊆ (𝑋 × V)
61	57, 60	sstrdi 4021	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V))
62		dfss2 3994	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
63	61, 62	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
64	55, 63	eqtr3id 2794	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I ))
65	54, 64	eqtrid 2792	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I ))
66	65	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅))
67		opelxp 5736	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑎 ∈ 𝑑))
68		df-br 5167	. . . . . . . . . . . . . . . 16 ⊢ (𝑎(𝑐 × 𝑑)𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑))
69		elin 3992	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (𝑐 ∩ 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑎 ∈ 𝑑))
70	67, 68, 69	3bitr4i 303	. . . . . . . . . . . . . . 15 ⊢ (𝑎(𝑐 × 𝑑)𝑎 ↔ 𝑎 ∈ (𝑐 ∩ 𝑑))
71	70	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐 ∩ 𝑑))
72	71	albii 1817	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐 ∩ 𝑑))
73		intirr 6150	. . . . . . . . . . . . 13 ⊢ (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎)
74		eq0 4373	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∩ 𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐 ∩ 𝑑))
75	72, 73, 74	3bitr4i 303	. . . . . . . . . . . 12 ⊢ (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅)
76	66, 75	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅))
77	52, 76	bitr3d 281	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑐 ∩ 𝑑) = ∅))
78	77	anbi2d 629	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 ∩ 𝑑) = ∅)))
79		opelxp 5736	. . . . . . . . . 10 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑))
80	79	anbi1i 623	. . . . . . . . 9 ⊢ ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))))
81		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 ∩ 𝑑) = ∅))
82	78, 80, 81	3bitr4g 314	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))
83	82	2rexbidva 3226	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))
84	41, 83	imbi12d 344	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))))
85	84	2ralbidva 3225	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))))
86	30, 85	bitrd 279	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑒 ∈ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪ (𝐽 ×_t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))))
87	11, 13, 86	3bitrrd 306	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)) ↔ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽))))
88	87	pm5.32i 574	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))) ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽))))
89	2, 88	bitri 275	1 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×_t 𝐽))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1535 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 ⟨cop 4654 ∪ cuni 4931 class class class wbr 5166 I cid 5592 × cxp 5698 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 Topctop 22920 Clsdccld 23045 Hauscha 23337 ×_t ctx 23589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-topgen 17503 df-top 22921 df-topon 22938 df-bases 22974 df-cld 23048 df-haus 23344 df-tx 23591

This theorem is referenced by: hauseqlcld 23675 tgphaus 24146 qtophaus 33782

W3C validator