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Theorem hausdiag 22996
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 22673 . 2 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
3 txtop 22920 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
43anidms 567 . . . . 5 (𝐽 ∈ Top → (𝐽 ×t 𝐽) ∈ Top)
5 idssxp 6002 . . . . . 6 ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋)
61, 1txuni 22943 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
76anidms 567 . . . . . 6 (𝐽 ∈ Top → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
85, 7sseqtrid 3996 . . . . 5 (𝐽 ∈ Top → ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽))
9 eqid 2736 . . . . . 6 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
109iscld2 22379 . . . . 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 22919 . . . . 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 3920 . . . . . . . . . 10 (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))
157eqcomd 2742 . . . . . . . . . . . 12 (𝐽 ∈ Top → (𝐽 ×t 𝐽) = (𝑋 × 𝑋))
1615eleq2d 2823 . . . . . . . . . . 11 (𝐽 ∈ Top → (𝑒 (𝐽 ×t 𝐽) ↔ 𝑒 ∈ (𝑋 × 𝑋)))
1716anbi1d 630 . . . . . . . . . 10 (𝐽 ∈ Top → ((𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1814, 17bitrid 282 . . . . . . . . 9 (𝐽 ∈ Top → (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
1918imbi1d 341 . . . . . . . 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, 20bitrdi 286 . . . . . . 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 3170 . . . . . 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 2825 . . . . . . . . 9 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ ( I ↾ 𝑋) ↔ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
2423notbid 317 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
25 eleq1 2825 . . . . . . . . . 10 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ (𝑐 × 𝑑) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑)))
2625anbi1d 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)))))
27262rexbidv 3213 . . . . . . . 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 344 . . . . . . 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 5797 . . . . . 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 286 . . . . 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 3449 . . . . . . . . . . 11 𝑏 ∈ V
3231opelresi 5945 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ))
33 ibar 529 . . . . . . . . . . . 12 (𝑎𝑋 → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
3433adantr 481 . . . . . . . . . . 11 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ I ↔ (𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I )))
35 df-br 5106 . . . . . . . . . . . 12 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
3631ideq 5808 . . . . . . . . . . . 12 (𝑎 I 𝑏𝑎 = 𝑏)
3735, 36bitr3i 276 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
3834, 37bitr3di 285 . . . . . . . . . 10 ((𝑎𝑋𝑏𝑋) → ((𝑎𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ I ) ↔ 𝑎 = 𝑏))
3932, 38bitrid 282 . . . . . . . . 9 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4039adantl 482 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4140necon3bbid 2981 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎𝑏))
42 elssuni 4898 . . . . . . . . . . . . . . . 16 (𝑐𝐽𝑐 𝐽)
43 elssuni 4898 . . . . . . . . . . . . . . . 16 (𝑑𝐽𝑑 𝐽)
44 xpss12 5648 . . . . . . . . . . . . . . . 16 ((𝑐 𝐽𝑑 𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
4542, 43, 44syl2an 596 . . . . . . . . . . . . . . 15 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
461, 1xpeq12i 5661 . . . . . . . . . . . . . . 15 (𝑋 × 𝑋) = ( 𝐽 × 𝐽)
4745, 46sseqtrrdi 3995 . . . . . . . . . . . . . 14 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
4847adantl 482 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
497ad2antrr 724 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
5048, 49sseqtrd 3984 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽))
51 reldisj 4411 . . . . . . . . . . . 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 5645 . . . . . . . . . . . . . . 15 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
5453ineq2i 4169 . . . . . . . . . . . . . 14 ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
55 inass 4179 . . . . . . . . . . . . . . 15 (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
56 inss1 4188 . . . . . . . . . . . . . . . . . 18 ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑)
5756, 48sstrid 3955 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋))
58 ssv 3968 . . . . . . . . . . . . . . . . . 18 𝑋 ⊆ V
59 xpss2 5653 . . . . . . . . . . . . . . . . . 18 (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V))
6058, 59ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑋 × 𝑋) ⊆ (𝑋 × V)
6157, 60sstrdi 3956 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V))
62 df-ss 3927 . . . . . . . . . . . . . . . 16 (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6361, 62sylib 217 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6455, 63eqtr3id 2790 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I ))
6554, 64eqtrid 2788 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I ))
6665eqeq1d 2738 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅))
67 opelxp 5669 . . . . . . . . . . . . . . . 16 (⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑎𝑑))
68 df-br 5106 . . . . . . . . . . . . . . . 16 (𝑎(𝑐 × 𝑑)𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑))
69 elin 3926 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (𝑐𝑑) ↔ (𝑎𝑐𝑎𝑑))
7067, 68, 693bitr4i 302 . . . . . . . . . . . . . . 15 (𝑎(𝑐 × 𝑑)𝑎𝑎 ∈ (𝑐𝑑))
7170notbii 319 . . . . . . . . . . . . . 14 𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐𝑑))
7271albii 1821 . . . . . . . . . . . . 13 (∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
73 intirr 6072 . . . . . . . . . . . . 13 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎)
74 eq0 4303 . . . . . . . . . . . . 13 ((𝑐𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
7572, 73, 743bitr4i 302 . . . . . . . . . . . 12 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐𝑑) = ∅)
7666, 75bitrdi 286 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐𝑑) = ∅))
7752, 76bitr3d 280 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑐𝑑) = ∅))
7877anbi2d 629 . . . . . . . . 9 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅)))
79 opelxp 5669 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑏𝑑))
8079anbi1i 624 . . . . . . . . 9 ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
81 df-3an 1089 . . . . . . . . 9 ((𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅))
8278, 80, 813bitr4g 313 . . . . . . . 8 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
83822rexbidva 3211 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
8441, 83imbi12d 344 . . . . . 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 3210 . . . . 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 278 . . . 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 305 . . 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 274 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  cop 4592   cuni 4865   class class class wbr 5105   I cid 5530   × cxp 5631  cres 5635  cfv 6496  (class class class)co 7357  Topctop 22242  Clsdccld 22367  Hauscha 22659   ×t ctx 22911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-haus 22666  df-tx 22913
This theorem is referenced by:  hauseqlcld  22997  tgphaus  23468  qtophaus  32417
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