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Theorem hausdiag 22335
 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 22012 . 2 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
3 txtop 22259 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
43anidms 571 . . . . 5 (𝐽 ∈ Top → (𝐽 ×t 𝐽) ∈ Top)
5 idssxp 5886 . . . . . 6 ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋)
61, 1txuni 22282 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
76anidms 571 . . . . . 6 (𝐽 ∈ Top → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
85, 7sseqtrid 3945 . . . . 5 (𝐽 ∈ Top → ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽))
9 eqid 2759 . . . . . 6 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
109iscld2 21718 . . . . 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 588 . . . 4 (𝐽 ∈ Top → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽)))
12 eltx 22258 . . . . 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 571 . . . 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 3869 . . . . . . . . . 10 (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))
157eqcomd 2765 . . . . . . . . . . . 12 (𝐽 ∈ Top → (𝐽 ×t 𝐽) = (𝑋 × 𝑋))
1615eleq2d 2838 . . . . . . . . . . 11 (𝐽 ∈ Top → (𝑒 (𝐽 ×t 𝐽) ↔ 𝑒 ∈ (𝑋 × 𝑋)))
1716anbi1d 633 . . . . . . . . . 10 (𝐽 ∈ Top → ((𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1814, 17syl5bb 286 . . . . . . . . 9 (𝐽 ∈ Top → (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1918imbi1d 346 . . . . . . . 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 455 . . . . . . . 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, 20bitrdi 290 . . . . . . 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 3125 . . . . . 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 2840 . . . . . . . . 9 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ ( I ↾ 𝑋) ↔ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
2423notbid 322 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
25 eleq1 2840 . . . . . . . . . 10 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ (𝑐 × 𝑑) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑)))
2625anbi1d 633 . . . . . . . . 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 3225 . . . . . . . 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 349 . . . . . . 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 5679 . . . . . 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, 29bitrdi 290 . . . . 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 3414 . . . . . . . . . . 11 𝑏 ∈ V
3231opelresi 5829 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ))
33 ibar 533 . . . . . . . . . . . 12 (𝑎𝑋 → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
3433adantr 485 . . . . . . . . . . 11 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
35 df-br 5031 . . . . . . . . . . . 12 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
3631ideq 5690 . . . . . . . . . . . 12 (𝑎 I 𝑏𝑎 = 𝑏)
3735, 36bitr3i 280 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
3834, 37bitr3di 289 . . . . . . . . . 10 ((𝑎𝑋𝑏𝑋) → ((𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ) ↔ 𝑎 = 𝑏))
3932, 38syl5bb 286 . . . . . . . . 9 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4039adantl 486 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4140necon3bbid 2989 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎𝑏))
42 elssuni 4828 . . . . . . . . . . . . . . . 16 (𝑐𝐽𝑐 𝐽)
43 elssuni 4828 . . . . . . . . . . . . . . . 16 (𝑑𝐽𝑑 𝐽)
44 xpss12 5537 . . . . . . . . . . . . . . . 16 ((𝑐 𝐽𝑑 𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
4542, 43, 44syl2an 599 . . . . . . . . . . . . . . 15 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
461, 1xpeq12i 5550 . . . . . . . . . . . . . . 15 (𝑋 × 𝑋) = ( 𝐽 × 𝐽)
4745, 46sseqtrrdi 3944 . . . . . . . . . . . . . 14 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
4847adantl 486 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
497ad2antrr 726 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
5048, 49sseqtrd 3933 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽))
51 reldisj 4346 . . . . . . . . . . . 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 5534 . . . . . . . . . . . . . . 15 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
5453ineq2i 4115 . . . . . . . . . . . . . 14 ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
55 inass 4125 . . . . . . . . . . . . . . 15 (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
56 inss1 4134 . . . . . . . . . . . . . . . . . 18 ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑)
5756, 48sstrid 3904 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋))
58 ssv 3917 . . . . . . . . . . . . . . . . . 18 𝑋 ⊆ V
59 xpss2 5542 . . . . . . . . . . . . . . . . . 18 (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V))
6058, 59ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑋 × 𝑋) ⊆ (𝑋 × V)
6157, 60sstrdi 3905 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V))
62 df-ss 3876 . . . . . . . . . . . . . . . 16 (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6361, 62sylib 221 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6455, 63syl5eqr 2808 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I ))
6554, 64syl5eq 2806 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I ))
6665eqeq1d 2761 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅))
67 opelxp 5558 . . . . . . . . . . . . . . . 16 (⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑎𝑑))
68 df-br 5031 . . . . . . . . . . . . . . . 16 (𝑎(𝑐 × 𝑑)𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑))
69 elin 3875 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (𝑐𝑑) ↔ (𝑎𝑐𝑎𝑑))
7067, 68, 693bitr4i 307 . . . . . . . . . . . . . . 15 (𝑎(𝑐 × 𝑑)𝑎𝑎 ∈ (𝑐𝑑))
7170notbii 324 . . . . . . . . . . . . . 14 𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐𝑑))
7271albii 1822 . . . . . . . . . . . . 13 (∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
73 intirr 5948 . . . . . . . . . . . . 13 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎)
74 eq0 4242 . . . . . . . . . . . . 13 ((𝑐𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
7572, 73, 743bitr4i 307 . . . . . . . . . . . 12 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐𝑑) = ∅)
7666, 75bitrdi 290 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐𝑑) = ∅))
7752, 76bitr3d 284 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑐𝑑) = ∅))
7877anbi2d 632 . . . . . . . . 9 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅)))
79 opelxp 5558 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑏𝑑))
8079anbi1i 627 . . . . . . . . 9 ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
81 df-3an 1087 . . . . . . . . 9 ((𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅))
8278, 80, 813bitr4g 318 . . . . . . . 8 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
83822rexbidva 3224 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
8441, 83imbi12d 349 . . . . . 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 3128 . . . . 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 282 . . . 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 310 . . 3 (𝐽 ∈ Top → (∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)) ↔ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
8887pm5.32i 579 . 2 ((𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))) ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
892, 88bitri 278 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  ⟨cop 4526  ∪ cuni 4796   class class class wbr 5030   I cid 5427   × cxp 5520   ↾ cres 5524  ‘cfv 6333  (class class class)co 7148  Topctop 21583  Clsdccld 21706  Hauscha 21998   ×t ctx 22250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7691  df-2nd 7692  df-topgen 16765  df-top 21584  df-topon 21601  df-bases 21636  df-cld 21709  df-haus 22005  df-tx 22252 This theorem is referenced by:  hauseqlcld  22336  tgphaus  22807  qtophaus  31297
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