Theorem haustop 22390

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 2738	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 22381	. 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 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  ∪ cuni 4836  Topctop 21950  Hauscha 22367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-haus 22374

This theorem is referenced by:  haust1  22411  resthaus  22427  sshaus  22434  lmmo  22439  hauscmplem  22465  hauscmp  22466  hauslly  22551  hausllycmp  22553  kgenhaus  22603  pthaus  22697  txhaus  22706  xkohaus  22712  haushmph  22851  cmphaushmeo  22859  hausflim  23040  hauspwpwf1  23046  hauspwpwdom  23047  hausflf  23056  cnextfun  23123  cnextfvval  23124  cnextf  23125  cnextcn  23126  cnextfres1  23127  cnextfres  23128  qtophaus  31688  ismntop  31876  poimirlem30  35734  hausgraph  40953