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Theorem ist1-2 22498
Description: An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ist1-2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑜,𝐽   𝑜,𝑋,𝑥,𝑦

Proof of Theorem ist1-2
StepHypRef Expression
1 topontop 22062 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2738 . . . . 5 𝐽 = 𝐽
32ist1 22472 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
43baib 536 . . 3 (𝐽 ∈ Top → (𝐽 ∈ Fre ↔ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
51, 4syl 17 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
6 toponuni 22063 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76raleqdv 3348 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
81adantr 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
9 eltop2 22125 . . . . . 6 (𝐽 ∈ Top → (( 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ ( 𝐽 ∖ {𝑦})∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
108, 9syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (( 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ ( 𝐽 ∖ {𝑦})∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
116eleq2d 2824 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑦𝑋𝑦 𝐽))
1211biimpa 477 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑦 𝐽)
1312snssd 4742 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → {𝑦} ⊆ 𝐽)
142iscld2 22179 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦} ⊆ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
158, 13, 14syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
166adantr 481 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑋 = 𝐽)
1716eleq2d 2824 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (𝑥𝑋𝑥 𝐽))
1817imbi1d 342 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))))
19 con1b 359 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20 df-ne 2944 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
2120imbi1i 350 . . . . . . . . 9 ((𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
22 disjsn 4647 . . . . . . . . . . . . . . 15 ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑜)
23 elssuni 4871 . . . . . . . . . . . . . . . 16 (𝑜𝐽𝑜 𝐽)
24 reldisj 4385 . . . . . . . . . . . . . . . 16 (𝑜 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑜𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2622, 25bitr3id 285 . . . . . . . . . . . . . 14 (𝑜𝐽 → (¬ 𝑦𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2726anbi2d 629 . . . . . . . . . . . . 13 (𝑜𝐽 → ((𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
2827rexbiia 3180 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
29 rexanali 3192 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3028, 29bitr3i 276 . . . . . . . . . . 11 (∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3130con2bii 358 . . . . . . . . . 10 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
3231imbi1i 350 . . . . . . . . 9 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
3319, 21, 323bitr4ri 304 . . . . . . . 8 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
3433imbi2i 336 . . . . . . 7 ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
35 eldifsn 4720 . . . . . . . . 9 (𝑥 ∈ ( 𝐽 ∖ {𝑦}) ↔ (𝑥 𝐽𝑥𝑦))
3635imbi1i 350 . . . . . . . 8 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ ((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
37 impexp 451 . . . . . . . 8 (((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3836, 37bitri 274 . . . . . . 7 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3918, 34, 383bitr4g 314 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
4039ralbidv2 3110 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ ( 𝐽 ∖ {𝑦})∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
4110, 15, 403bitr4d 311 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
4241ralbidva 3111 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
43 ralcom 3166 . . 3 (∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
4442, 43bitrdi 287 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
455, 7, 443bitr2d 307 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  cdif 3884  cin 3886  wss 3887  c0 4256  {csn 4561   cuni 4839  cfv 6433  Topctop 22042  TopOnctopon 22059  Clsdccld 22167  Frect1 22458
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154  df-top 22043  df-topon 22060  df-cld 22170  df-t1 22465
This theorem is referenced by:  t1t0  22499  ist1-3  22500  haust1  22503  t1sep2  22520  isr0  22888  tgpt0  23270
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