Theorem haustop 23251

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 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 23242	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∩ cin 3910  ∅c0 4292  ∪ cuni 4867  Topctop 22813  Hauscha 23228

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-ss 3928  df-uni 4868  df-haus 23235

This theorem is referenced by:  haust1  23272  resthaus  23288  sshaus  23295  lmmo  23300  hauscmplem  23326  hauscmp  23327  hauslly  23412  hausllycmp  23414  kgenhaus  23464  pthaus  23558  txhaus  23567  xkohaus  23573  haushmph  23712  cmphaushmeo  23720  hausflim  23901  hauspwpwf1  23907  hauspwpwdom  23908  hausflf  23917  cnextfun  23984  cnextfvval  23985  cnextf  23986  cnextcn  23987  cnextfres1  23988  cnextfres  23989  qtophaus  33819  ismntop  34009  poimirlem30  37637  hausgraph  43187