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 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))

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.

Step	Hyp	Ref	Expression
1		ist0.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
2	1	ishaus 20936	. . . . . 6 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simprbi 479	. . . . 5 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
4		neeq1 2844	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
5		eleq1 2676	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
6	5	3anbi1d 1395	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
7	6	2rexbidv 3039	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
8	4, 7	imbi12d 333	. . . . . 6 ⊢ (𝑥 = 𝑃 → ((𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) ↔ (𝑃 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
9		neeq2 2845	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
10		eleq1 2676	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑦 ∈ 𝑚 ↔ 𝑄 ∈ 𝑚))
11	10	3anbi2d 1396	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → ((𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
12	11	2rexbidv 3039	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
13	9, 12	imbi12d 333	. . . . . 6 ⊢ (𝑦 = 𝑄 → ((𝑃 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) ↔ (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
14	8, 13	rspc2v 3293	. . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
15	3, 14	syl5 33	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝐽 ∈ Haus → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
16	15	ex 449	. . 3 ⊢ (𝑃 ∈ 𝑋 → (𝑄 ∈ 𝑋 → (𝐽 ∈ Haus → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
17	16	com3r 85	. 2 ⊢ (𝐽 ∈ Haus → (𝑃 ∈ 𝑋 → (𝑄 ∈ 𝑋 → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
18	17	3imp2 1274	1 ⊢ ((𝐽 ∈ Haus ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))

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