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Theorem hausnei 21461
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 21455 . . . . . 6 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
32simprbi 491 . . . . 5 (𝐽 ∈ Haus → ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
4 neeq1 3033 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
5 eleq1 2866 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥𝑛𝑃𝑛))
653anbi1d 1565 . . . . . . . 8 (𝑥 = 𝑃 → ((𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
762rexbidv 3238 . . . . . . 7 (𝑥 = 𝑃 → (∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
84, 7imbi12d 336 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 neeq2 3034 . . . . . . 7 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
10 eleq1 2866 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑦𝑚𝑄𝑚))
11103anbi2d 1566 . . . . . . . 8 (𝑦 = 𝑄 → ((𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
12112rexbidv 3238 . . . . . . 7 (𝑦 = 𝑄 → (∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
139, 12imbi12d 336 . . . . . 6 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
148, 13rspc2v 3510 . . . . 5 ((𝑃𝑋𝑄𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
153, 14syl5 34 . . . 4 ((𝑃𝑋𝑄𝑋) → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
1615ex 402 . . 3 (𝑃𝑋 → (𝑄𝑋 → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
1716com3r 87 . 2 (𝐽 ∈ Haus → (𝑃𝑋 → (𝑄𝑋 → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
18173imp2 1459 1 ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090  cin 3768  c0 4115   cuni 4628  Topctop 21026  Hauscha 21441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-uni 4629  df-haus 21448
This theorem is referenced by:  haust1  21485  cnhaus  21487  lmmo  21513  hauscmplem  21538  pthaus  21770  txhaus  21779  xkohaus  21785  hausflimi  22112  hauspwpwf1  22119
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