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Theorem hausdiag 22169
 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.
StepHypRef Expression
1 hausdiag.x . . 3 𝑋 = 𝐽
21ishaus 21846 . 2 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
3 txtop 22093 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
43anidms 567 . . . . 5 (𝐽 ∈ Top → (𝐽 ×t 𝐽) ∈ Top)
5 idssxp 5915 . . . . . 6 ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋)
61, 1txuni 22116 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
76anidms 567 . . . . . 6 (𝐽 ∈ Top → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
85, 7sseqtrid 4023 . . . . 5 (𝐽 ∈ Top → ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽))
9 eqid 2826 . . . . . 6 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
109iscld2 21552 . . . . 5 (((𝐽 ×t 𝐽) ∈ Top ∧ ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽)) → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽)))
114, 8, 10syl2anc 584 . . . 4 (𝐽 ∈ Top → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽)))
12 eltx 22092 . . . . 5 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽) ↔ ∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
1312anidms 567 . . . 4 (𝐽 ∈ Top → (( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽) ↔ ∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
14 eldif 3950 . . . . . . . . . 10 (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))
157eqcomd 2832 . . . . . . . . . . . 12 (𝐽 ∈ Top → (𝐽 ×t 𝐽) = (𝑋 × 𝑋))
1615eleq2d 2903 . . . . . . . . . . 11 (𝐽 ∈ Top → (𝑒 (𝐽 ×t 𝐽) ↔ 𝑒 ∈ (𝑋 × 𝑋)))
1716anbi1d 629 . . . . . . . . . 10 (𝐽 ∈ Top → ((𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1814, 17syl5bb 284 . . . . . . . . 9 (𝐽 ∈ Top → (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1918imbi1d 343 . . . . . . . 8 (𝐽 ∈ Top → ((𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
20 impexp 451 . . . . . . . 8 (((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
2119, 20syl6bb 288 . . . . . . 7 (𝐽 ∈ Top → ((𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))))
2221ralbidv2 3200 . . . . . 6 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
23 eleq1 2905 . . . . . . . . 9 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ ( I ↾ 𝑋) ↔ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
2423notbid 319 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
25 eleq1 2905 . . . . . . . . . 10 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ (𝑐 × 𝑑) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑)))
2625anbi1d 629 . . . . . . . . 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)))))
27262rexbidv 3305 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
2824, 27imbi12d 346 . . . . . . 7 (𝑒 = ⟨𝑎, 𝑏⟩ → ((¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
2928ralxp 5711 . . . . . 6 (∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
3022, 29syl6bb 288 . . . . 5 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
31 vex 3503 . . . . . . . . . . 11 𝑏 ∈ V
3231opelresi 5860 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ))
33 df-br 5064 . . . . . . . . . . . 12 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
3431ideq 5722 . . . . . . . . . . . 12 (𝑎 I 𝑏𝑎 = 𝑏)
3533, 34bitr3i 278 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
36 ibar 529 . . . . . . . . . . . 12 (𝑎𝑋 → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
3736adantr 481 . . . . . . . . . . 11 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
3835, 37syl5rbbr 287 . . . . . . . . . 10 ((𝑎𝑋𝑏𝑋) → ((𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ) ↔ 𝑎 = 𝑏))
3932, 38syl5bb 284 . . . . . . . . 9 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4039adantl 482 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4140necon3bbid 3058 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎𝑏))
42 elssuni 4866 . . . . . . . . . . . . . . . 16 (𝑐𝐽𝑐 𝐽)
43 elssuni 4866 . . . . . . . . . . . . . . . 16 (𝑑𝐽𝑑 𝐽)
44 xpss12 5569 . . . . . . . . . . . . . . . 16 ((𝑐 𝐽𝑑 𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
4542, 43, 44syl2an 595 . . . . . . . . . . . . . . 15 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
461, 1xpeq12i 5582 . . . . . . . . . . . . . . 15 (𝑋 × 𝑋) = ( 𝐽 × 𝐽)
4745, 46sseqtrrdi 4022 . . . . . . . . . . . . . 14 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
4847adantl 482 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
497ad2antrr 722 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
5048, 49sseqtrd 4011 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽))
51 reldisj 4405 . . . . . . . . . . . 12 ((𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
5250, 51syl 17 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
53 df-res 5566 . . . . . . . . . . . . . . 15 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
5453ineq2i 4190 . . . . . . . . . . . . . 14 ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
55 inass 4200 . . . . . . . . . . . . . . 15 (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
56 inss1 4209 . . . . . . . . . . . . . . . . . 18 ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑)
5756, 48sstrid 3982 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋))
58 ssv 3995 . . . . . . . . . . . . . . . . . 18 𝑋 ⊆ V
59 xpss2 5574 . . . . . . . . . . . . . . . . . 18 (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V))
6058, 59ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑋 × 𝑋) ⊆ (𝑋 × V)
6157, 60syl6ss 3983 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V))
62 df-ss 3956 . . . . . . . . . . . . . . . 16 (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6361, 62sylib 219 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6455, 63syl5eqr 2875 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I ))
6554, 64syl5eq 2873 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I ))
6665eqeq1d 2828 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅))
67 opelxp 5590 . . . . . . . . . . . . . . . 16 (⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑎𝑑))
68 df-br 5064 . . . . . . . . . . . . . . . 16 (𝑎(𝑐 × 𝑑)𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑))
69 elin 4173 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (𝑐𝑑) ↔ (𝑎𝑐𝑎𝑑))
7067, 68, 693bitr4i 304 . . . . . . . . . . . . . . 15 (𝑎(𝑐 × 𝑑)𝑎𝑎 ∈ (𝑐𝑑))
7170notbii 321 . . . . . . . . . . . . . 14 𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐𝑑))
7271albii 1813 . . . . . . . . . . . . 13 (∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
73 intirr 5976 . . . . . . . . . . . . 13 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎)
74 eq0 4312 . . . . . . . . . . . . 13 ((𝑐𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
7572, 73, 743bitr4i 304 . . . . . . . . . . . 12 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐𝑑) = ∅)
7666, 75syl6bb 288 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐𝑑) = ∅))
7752, 76bitr3d 282 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑐𝑑) = ∅))
7877anbi2d 628 . . . . . . . . 9 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅)))
79 opelxp 5590 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑏𝑑))
8079anbi1i 623 . . . . . . . . 9 ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
81 df-3an 1083 . . . . . . . . 9 ((𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅))
8278, 80, 813bitr4g 315 . . . . . . . 8 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
83822rexbidva 3304 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
8441, 83imbi12d 346 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → ((¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
85842ralbidva 3203 . . . . 5 (𝐽 ∈ Top → (∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
8630, 85bitrd 280 . . . 4 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
8711, 13, 863bitrrd 307 . . 3 (𝐽 ∈ Top → (∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)) ↔ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
8887pm5.32i 575 . 2 ((𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))) ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
892, 88bitri 276 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081  ∀wal 1528   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∖ cdif 3937   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063   I cid 5458   × cxp 5552   ↾ cres 5556  ‘cfv 6352  (class class class)co 7148  Topctop 21417  Clsdccld 21540  Hauscha 21832   ×t ctx 22084 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-topgen 16707  df-top 21418  df-topon 21435  df-bases 21470  df-cld 21543  df-haus 21839  df-tx 22086 This theorem is referenced by:  hauseqlcld  22170  tgphaus  22640  qtophaus  30986
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