Theorem haustop 23055

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 2730	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 23046	. 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 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ∩ cin 3946  ∅c0 4321  ∪ cuni 4907  Topctop 22615  Hauscha 23032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-haus 23039

This theorem is referenced by:  haust1  23076  resthaus  23092  sshaus  23099  lmmo  23104  hauscmplem  23130  hauscmp  23131  hauslly  23216  hausllycmp  23218  kgenhaus  23268  pthaus  23362  txhaus  23371  xkohaus  23377  haushmph  23516  cmphaushmeo  23524  hausflim  23705  hauspwpwf1  23711  hauspwpwdom  23712  hausflf  23721  cnextfun  23788  cnextfvval  23789  cnextf  23790  cnextcn  23791  cnextfres1  23792  cnextfres  23793  qtophaus  33114  ismntop  33304  poimirlem30  36821  hausgraph  42256