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Theorem hausnei 23052
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 23046 . . . . . 6 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
32simprbi 495 . . . . 5 (𝐽 ∈ Haus → ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
4 neeq1 3001 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
5 eleq1 2819 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥𝑛𝑃𝑛))
653anbi1d 1438 . . . . . . . 8 (𝑥 = 𝑃 → ((𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
762rexbidv 3217 . . . . . . 7 (𝑥 = 𝑃 → (∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
84, 7imbi12d 343 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 neeq2 3002 . . . . . . 7 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
10 eleq1 2819 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑦𝑚𝑄𝑚))
11103anbi2d 1439 . . . . . . . 8 (𝑦 = 𝑄 → ((𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
12112rexbidv 3217 . . . . . . 7 (𝑦 = 𝑄 → (∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
139, 12imbi12d 343 . . . . . 6 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
148, 13rspc2v 3621 . . . . 5 ((𝑃𝑋𝑄𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
153, 14syl5 34 . . . 4 ((𝑃𝑋𝑄𝑋) → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
1615ex 411 . . 3 (𝑃𝑋 → (𝑄𝑋 → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
1716com3r 87 . 2 (𝐽 ∈ Haus → (𝑃𝑋 → (𝑄𝑋 → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
18173imp2 1347 1 ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  cin 3946  c0 4321   cuni 4907  Topctop 22615  Hauscha 23032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-haus 23039
This theorem is referenced by:  haust1  23076  cnhaus  23078  lmmo  23104  hauscmplem  23130  pthaus  23362  txhaus  23371  xkohaus  23377  hausflimi  23704  hauspwpwf1  23711
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