Theorem haustop 23499

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 2762	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 23490	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ∩ cin 3903  ∅c0 4285  ∪ cuni 4871  Topctop 23061  Hauscha 23476

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-ss 3921  df-uni 4872  df-haus 23483

This theorem is used by:  haust1  23520  resthaus  23536  sshaus  23543  lmmo  23548  hauscmplem  23574  hauscmp  23575  hauslly  23660  hausllycmp  23662  kgenhaus  23712  pthaus  23806  txhaus  23815  xkohaus  23821  haushmph  23960  cmphaushmeo  23968  hausflim  24149  hauspwpwf1  24155  hauspwpwdom  24156  hausflf  24165  cnextfun  24232  cnextfvval  24233  cnextf  24234  cnextcn  24235  cnextfres1  24236  cnextfres  24237  qtophaus  34235  ismntop  34425  poimirlem30  38329  hausgraph  43960