Theorem haustop 23296

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 2736	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 23287	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∩ cin 3888  ∅c0 4273  ∪ cuni 4850  Topctop 22858  Hauscha 23273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-ss 3906  df-uni 4851  df-haus 23280

This theorem is referenced by:  haust1  23317  resthaus  23333  sshaus  23340  lmmo  23345  hauscmplem  23371  hauscmp  23372  hauslly  23457  hausllycmp  23459  kgenhaus  23509  pthaus  23603  txhaus  23612  xkohaus  23618  haushmph  23757  cmphaushmeo  23765  hausflim  23946  hauspwpwf1  23952  hauspwpwdom  23953  hausflf  23962  cnextfun  24029  cnextfvval  24030  cnextf  24031  cnextcn  24032  cnextfres1  24033  cnextfres  24034  qtophaus  33980  ismntop  34170  poimirlem30  37971  hausgraph  43633