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Theorem haust1 21961
Description: A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
haust1 (𝐽 ∈ Haus → 𝐽 ∈ Fre)

Proof of Theorem haust1
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . . . . . . 9 𝐽 = 𝐽
21hausnei 21937 . . . . . . . 8 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑧𝐽𝑤𝐽 (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
3 simprr1 1218 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → 𝑥𝑧)
4 noel 4250 . . . . . . . . . . . . 13 ¬ 𝑦 ∈ ∅
5 simprr3 1220 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → (𝑧𝑤) = ∅)
65eleq2d 2878 . . . . . . . . . . . . 13 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → (𝑦 ∈ (𝑧𝑤) ↔ 𝑦 ∈ ∅))
74, 6mtbiri 330 . . . . . . . . . . . 12 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → ¬ 𝑦 ∈ (𝑧𝑤))
8 simprr2 1219 . . . . . . . . . . . . 13 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → 𝑦𝑤)
9 elin 3900 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑧𝑤) ↔ (𝑦𝑧𝑦𝑤))
109simplbi2com 506 . . . . . . . . . . . . 13 (𝑦𝑤 → (𝑦𝑧𝑦 ∈ (𝑧𝑤)))
118, 10syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → (𝑦𝑧𝑦 ∈ (𝑧𝑤)))
127, 11mtod 201 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → ¬ 𝑦𝑧)
133, 12jca 515 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) ∧ (𝑤𝐽 ∧ (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))) → (𝑥𝑧 ∧ ¬ 𝑦𝑧))
1413rexlimdvaa 3247 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) ∧ 𝑧𝐽) → (∃𝑤𝐽 (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅) → (𝑥𝑧 ∧ ¬ 𝑦𝑧)))
1514reximdva 3236 . . . . . . . 8 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → (∃𝑧𝐽𝑤𝐽 (𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅) → ∃𝑧𝐽 (𝑥𝑧 ∧ ¬ 𝑦𝑧)))
162, 15mpd 15 . . . . . . 7 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ¬ 𝑦𝑧))
17 rexanali 3227 . . . . . . 7 (∃𝑧𝐽 (𝑥𝑧 ∧ ¬ 𝑦𝑧) ↔ ¬ ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
1816, 17sylib 221 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ¬ ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
19183exp2 1351 . . . . 5 (𝐽 ∈ Haus → (𝑥 𝐽 → (𝑦 𝐽 → (𝑥𝑦 → ¬ ∀𝑧𝐽 (𝑥𝑧𝑦𝑧)))))
2019imp32 422 . . . 4 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽)) → (𝑥𝑦 → ¬ ∀𝑧𝐽 (𝑥𝑧𝑦𝑧)))
2120necon4ad 3009 . . 3 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽)) → (∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦))
2221ralrimivva 3159 . 2 (𝐽 ∈ Haus → ∀𝑥 𝐽𝑦 𝐽(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦))
23 haustop 21940 . . . 4 (𝐽 ∈ Haus → 𝐽 ∈ Top)
24 toptopon2 21527 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2523, 24sylib 221 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘ 𝐽))
26 ist1-2 21956 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
2725, 26syl 17 . 2 (𝐽 ∈ Haus → (𝐽 ∈ Fre ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
2822, 27mpbird 260 1 (𝐽 ∈ Haus → 𝐽 ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  cin 3883  c0 4246   cuni 4803  cfv 6328  Topctop 21502  TopOnctopon 21519  Frect1 21916  Hauscha 21917
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-topgen 16713  df-top 21503  df-topon 21520  df-cld 21628  df-t1 21923  df-haus 21924
This theorem is referenced by:  sncld  21980  ishaus3  22432  reghaus  22434  nrmhaus  22435  tgpt1  22727  metreg  23472  ipasslem8  28624  sitmcl  31723  onint1  33911  oninhaus  33912  poimirlem30  35086
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