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Theorem hausnei 20942
 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 20936 . . . . . 6 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
32simprbi 479 . . . . 5 (𝐽 ∈ Haus → ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
4 neeq1 2844 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
5 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥𝑛𝑃𝑛))
653anbi1d 1395 . . . . . . . 8 (𝑥 = 𝑃 → ((𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
762rexbidv 3039 . . . . . . 7 (𝑥 = 𝑃 → (∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
84, 7imbi12d 333 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 neeq2 2845 . . . . . . 7 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
10 eleq1 2676 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑦𝑚𝑄𝑚))
11103anbi2d 1396 . . . . . . . 8 (𝑦 = 𝑄 → ((𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
12112rexbidv 3039 . . . . . . 7 (𝑦 = 𝑄 → (∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))
139, 12imbi12d 333 . . . . . 6 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
148, 13rspc2v 3293 . . . . 5 ((𝑃𝑋𝑄𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
153, 14syl5 33 . . . 4 ((𝑃𝑋𝑄𝑋) → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))))
1615ex 449 . . 3 (𝑃𝑋 → (𝑄𝑋 → (𝐽 ∈ Haus → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
1716com3r 85 . 2 (𝐽 ∈ Haus → (𝑃𝑋 → (𝑄𝑋 → (𝑃𝑄 → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅)))))
18173imp2 1274 1 ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  ∅c0 3874  ∪ cuni 4372  Topctop 20517  Hauscha 20922 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-uni 4373  df-haus 20929 This theorem is referenced by:  haust1  20966  cnhaus  20968  lmmo  20994  hauscmplem  21019  pthaus  21251  txhaus  21260  xkohaus  21266  hausflimi  21594  hauspwpwf1  21601
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