Theorem haustop 23676

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 2763	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 23667	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ∩ cin 4092  ∅c0 4281  ∪ cuni 4870  Topctop 23238  Hauscha 23653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-uni 4871  df-haus 23660

This theorem is used by:  haust1  23697  resthaus  23713  sshaus  23720  lmmo  23725  hauscmplem  23751  hauscmp  23752  hauslly  23838  hausllycmp  23840  kgenhaus  23890  pthaus  23984  txhaus  23993  xkohaus  23999  haushmph  24138  cmphaushmeo  24146  hausflim  24327  hauspwpwf1  24333  hauspwpwdom  24334  hausflf  24343  cnextfun  24410  cnextfvval  24411  cnextf  24412  cnextcn  24413  cnextfres1  24414  cnextfres  24415  qtophaus  34535  ismntop  34725  poimirlem30  38759  hausgraph  44397