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Theorem ist1-2 21431
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 20997 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2765 . . . . 5 𝐽 = 𝐽
32ist1 21405 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
43baib 531 . . 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 20998 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76raleqdv 3292 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 𝐽{𝑦} ∈ (Clsd‘𝐽)))
81adantr 472 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
9 eltop2 21059 . . . . . 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 2830 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑦𝑋𝑦 𝐽))
1211biimpa 468 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑦 𝐽)
1312snssd 4494 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → {𝑦} ⊆ 𝐽)
142iscld2 21112 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦} ⊆ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
158, 13, 14syl2anc 579 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ {𝑦}) ∈ 𝐽))
166adantr 472 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑋 = 𝐽)
1716eleq2d 2830 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (𝑥𝑋𝑥 𝐽))
1817imbi1d 332 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))))
19 con1b 349 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20 df-ne 2938 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
2120imbi1i 340 . . . . . . . . 9 ((𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
22 disjsn 4402 . . . . . . . . . . . . . . 15 ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑜)
23 elssuni 4625 . . . . . . . . . . . . . . . 16 (𝑜𝐽𝑜 𝐽)
24 reldisj 4181 . . . . . . . . . . . . . . . 16 (𝑜 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑜𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2622, 25syl5bbr 276 . . . . . . . . . . . . . 14 (𝑜𝐽 → (¬ 𝑦𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
2726anbi2d 622 . . . . . . . . . . . . 13 (𝑜𝐽 → ((𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
2827rexbiia 3187 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
29 rexanali 3144 . . . . . . . . . . . 12 (∃𝑜𝐽 (𝑥𝑜 ∧ ¬ 𝑦𝑜) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3028, 29bitr3i 268 . . . . . . . . . . 11 (∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
3130con2bii 348 . . . . . . . . . 10 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))
3231imbi1i 340 . . . . . . . . 9 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
3319, 21, 323bitr4ri 295 . . . . . . . 8 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
3433imbi2i 327 . . . . . . 7 ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥𝑋 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
35 eldifsn 4472 . . . . . . . . 9 (𝑥 ∈ ( 𝐽 ∖ {𝑦}) ↔ (𝑥 𝐽𝑥𝑦))
3635imbi1i 340 . . . . . . . 8 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ ((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
37 impexp 441 . . . . . . . 8 (((𝑥 𝐽𝑥𝑦) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3836, 37bitri 266 . . . . . . 7 ((𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))) ↔ (𝑥 𝐽 → (𝑥𝑦 → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
3918, 34, 383bitr4g 305 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ((𝑥𝑋 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ ( 𝐽 ∖ {𝑦}) → ∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦})))))
4039ralbidv2 3131 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → (∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ ( 𝐽 ∖ {𝑦})∃𝑜𝐽 (𝑥𝑜𝑜 ⊆ ( 𝐽 ∖ {𝑦}))))
4110, 15, 403bitr4d 302 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
4241ralbidva 3132 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
43 ralcom 3245 . . 3 (∀𝑦𝑋𝑥𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
4442, 43syl6bb 278 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
455, 7, 443bitr2d 298 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  cdif 3729  cin 3731  wss 3732  c0 4079  {csn 4334   cuni 4594  cfv 6068  Topctop 20977  TopOnctopon 20994  Clsdccld 21100  Frect1 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-topgen 16372  df-top 20978  df-topon 20995  df-cld 21103  df-t1 21398
This theorem is referenced by:  t1t0  21432  ist1-3  21433  haust1  21436  t1sep2  21453  isr0  21820  tgpt0  22201
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