Theorem haustop 22554

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 22545	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 498	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ∩ cin 3896  ∅c0 4267  ∪ cuni 4850  Topctop 22114  Hauscha 22531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-in 3904  df-ss 3914  df-uni 4851  df-haus 22538

This theorem is referenced by:  haust1  22575  resthaus  22591  sshaus  22598  lmmo  22603  hauscmplem  22629  hauscmp  22630  hauslly  22715  hausllycmp  22717  kgenhaus  22767  pthaus  22861  txhaus  22870  xkohaus  22876  haushmph  23015  cmphaushmeo  23023  hausflim  23204  hauspwpwf1  23210  hauspwpwdom  23211  hausflf  23220  cnextfun  23287  cnextfvval  23288  cnextf  23289  cnextcn  23290  cnextfres1  23291  cnextfres  23292  qtophaus  31892  ismntop  32082  poimirlem30  35863  hausgraph  41241