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Theorem ist1-2 21056
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 20636 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2626 . . . . 5 𝐽 = 𝐽
32ist1 21030 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
43baib 943 . . 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 20637 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76raleqdv 3138 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
81adantr 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
9 eltop2 20685 . . . . . 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 2689 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑦𝑋𝑦 𝐽))
1211biimpa 501 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑦 𝐽)
1312snssd 4314 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → {𝑦} ⊆ 𝐽)
142iscld2 20737 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦} ⊆ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
158, 13, 14syl2anc 692 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
166adantr 481 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑋 = 𝐽)
1716eleq2d 2689 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (𝑥𝑋𝑥 𝐽))
1817imbi1d 331 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))))
19 con1b 348 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20 df-ne 2797 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
2120imbi1i 339 . . . . . . . . 9 ((𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
22 disjsn 4221 . . . . . . . . . . . . . . 15 ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑜)
23 elssuni 4438 . . . . . . . . . . . . . . . 16 (𝑜𝐽𝑜 𝐽)
24 reldisj 3997 . . . . . . . . . . . . . . . 16 (𝑜 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑜𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2622, 25syl5bbr 274 . . . . . . . . . . . . . 14 (𝑜𝐽 → (¬ 𝑦𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2726anbi2d 739 . . . . . . . . . . . . 13 (𝑜𝐽 → ((𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
2827rexbiia 3038 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
29 rexanali 2997 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3028, 29bitr3i 266 . . . . . . . . . . 11 (∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3130con2bii 347 . . . . . . . . . 10 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
3231imbi1i 339 . . . . . . . . 9 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
3319, 21, 323bitr4ri 293 . . . . . . . 8 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
3433imbi2i 326 . . . . . . 7 ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
35 eldifsn 4292 . . . . . . . . 9 (𝑥 ∈ ( 𝐽 ∖ {𝑦}) ↔ (𝑥 𝐽𝑥𝑦))
3635imbi1i 339 . . . . . . . 8 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ ((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
37 impexp 462 . . . . . . . 8 (((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3836, 37bitri 264 . . . . . . 7 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3918, 34, 383bitr4g 303 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
4039ralbidv2 2983 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ ( 𝐽 ∖ {𝑦})∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
4110, 15, 403bitr4d 300 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
4241ralbidva 2984 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
43 ralcom 3095 . . 3 (∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
4442, 43syl6bb 276 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
455, 7, 443bitr2d 296 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cdif 3557  cin 3559  wss 3560  c0 3896  {csn 4153   cuni 4407  cfv 5850  Topctop 20612  TopOnctopon 20613  Clsdccld 20725  Frect1 21016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-topgen 16020  df-top 20616  df-topon 20618  df-cld 20728  df-t1 21023
This theorem is referenced by:  t1t0  21057  ist1-3  21058  haust1  21061  t1sep2  21078  isr0  21445  tgpt0  21827
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