Theorem hausnei 22479

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 22473	. . . . . 6 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simprbi 497	. . . . 5 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
4		neeq1 3006	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
5		eleq1 2826	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
6	5	3anbi1d 1439	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
7	6	2rexbidv 3229	. . . . . . 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 345	. . . . . 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 3007	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
10		eleq1 2826	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑦 ∈ 𝑚 ↔ 𝑄 ∈ 𝑚))
11	10	3anbi2d 1440	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → ((𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
12	11	2rexbidv 3229	. . . . . . 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 345	. . . . . 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 3570	. . . . 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 34	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝐽 ∈ Haus → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
16	15	ex 413	. . 3 ⊢ (𝑃 ∈ 𝑋 → (𝑄 ∈ 𝑋 → (𝐽 ∈ Haus → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
17	16	com3r 87	. 2 ⊢ (𝐽 ∈ Haus → (𝑃 ∈ 𝑋 → (𝑄 ∈ 𝑋 → (𝑃 ≠ 𝑄 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
18	17	3imp2 1348	1 ⊢ ((𝐽 ∈ Haus ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886  ∅c0 4256  ∪ cuni 4839  Topctop 22042  Hauscha 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-haus 22466

This theorem is referenced by:  haust1  22503  cnhaus  22505  lmmo  22531  hauscmplem  22557  pthaus  22789  txhaus  22798  xkohaus  22804  hausflimi  23131  hauspwpwf1  23138