Theorem haustop 22228

Description: A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)

Assertion

Ref	Expression
haustop	⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Proof of Theorem haustop

Dummy variables 𝑥 𝑦 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 22219	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062   ∩ cin 3865  ∅c0 4237  ∪ cuni 4819  Topctop 21790  Hauscha 22205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-haus 22212

This theorem is referenced by:  haust1  22249  resthaus  22265  sshaus  22272  lmmo  22277  hauscmplem  22303  hauscmp  22304  hauslly  22389  hausllycmp  22391  kgenhaus  22441  pthaus  22535  txhaus  22544  xkohaus  22550  haushmph  22689  cmphaushmeo  22697  hausflim  22878  hauspwpwf1  22884  hauspwpwdom  22885  hausflf  22894  cnextfun  22961  cnextfvval  22962  cnextf  22963  cnextcn  22964  cnextfres1  22965  cnextfres  22966  qtophaus  31500  ismntop  31688  poimirlem30  35544  hausgraph  40740