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Theorem hausnei 21072
Description: Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
hausnei ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
Distinct variable groups:   𝑚,𝑛,𝐽   𝑃,𝑚,𝑛   𝑄,𝑚,𝑛
Allowed substitution hints:   𝑋(𝑚,𝑛)

Proof of Theorem hausnei
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . . . . 7 𝑋 = 𝐽
21ishaus 21066 . . . . . 6 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
32simprbi 480 . . . . 5 (𝐽 ∈ Haus → ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
4 neeq1 2852 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
5 eleq1 2686 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥𝑛𝑃𝑛))
653anbi1d 1400 . . . . . . . 8 (𝑥 = 𝑃 → ((𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
762rexbidv 3052 . . . . . . 7 (𝑥 = 𝑃 → (∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
84, 7imbi12d 334 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 neeq2 2853 . . . . . . 7 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
10 eleq1 2686 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑦𝑚𝑄𝑚))
11103anbi2d 1401 . . . . . . . 8 (𝑦 = 𝑄 → ((𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
12112rexbidv 3052 . . . . . . 7 (𝑦 = 𝑄 → (∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
139, 12imbi12d 334 . . . . . 6 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
148, 13rspc2v 3311 . . . . 5 ((𝑃𝑋𝑄𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
153, 14syl5 34 . . . 4 ((𝑃𝑋𝑄𝑋) → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
1615ex 450 . . 3 (𝑃𝑋 → (𝑄𝑋 → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
1716com3r 87 . 2 (𝐽 ∈ Haus → (𝑃𝑋 → (𝑄𝑋 → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
18173imp2 1279 1 ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  cin 3559  c0 3897   cuni 4409  Topctop 20638  Hauscha 21052
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-uni 4410  df-haus 21059
This theorem is referenced by:  haust1  21096  cnhaus  21098  lmmo  21124  hauscmplem  21149  pthaus  21381  txhaus  21390  xkohaus  21396  hausflimi  21724  hauspwpwf1  21731
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