Theorem haustop 21933

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 2821	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 21924	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 500	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∩ cin 3935  ∅c0 4291  ∪ cuni 4832  Topctop 21495  Hauscha 21910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-uni 4833  df-haus 21917

This theorem is referenced by:  haust1  21954  resthaus  21970  sshaus  21977  lmmo  21982  hauscmplem  22008  hauscmp  22009  hauslly  22094  hausllycmp  22096  kgenhaus  22146  pthaus  22240  txhaus  22249  xkohaus  22255  haushmph  22394  cmphaushmeo  22402  hausflim  22583  hauspwpwf1  22589  hauspwpwdom  22590  hausflf  22599  cnextfun  22666  cnextfvval  22667  cnextf  22668  cnextcn  22669  cnextfres1  22670  cnextfres  22671  qtophaus  31095  ismntop  31262  poimirlem30  34916  hausgraph  39805