Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  haustop Structured version   Visualization version   GIF version

Theorem haustop 21045
 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.
StepHypRef Expression
1 eqid 2621 . . 3 𝐽 = 𝐽
21ishaus 21036 . 2 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
32simplbi 476 1 (𝐽 ∈ Haus → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ∩ cin 3554  ∅c0 3891  ∪ cuni 4402  Topctop 20617  Hauscha 21022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-uni 4403  df-haus 21029 This theorem is referenced by:  haust1  21066  resthaus  21082  sshaus  21089  lmmo  21094  hauscmplem  21119  hauscmp  21120  hauslly  21205  hausllycmp  21207  kgenhaus  21257  pthaus  21351  txhaus  21360  xkohaus  21366  haushmph  21505  cmphaushmeo  21513  hausflim  21695  hauspwpwf1  21701  hauspwpwdom  21702  hausflf  21711  cnextfun  21778  cnextfvval  21779  cnextf  21780  cnextcn  21781  cnextfres1  21782  cnextfres  21783  qtophaus  29685  ismntop  29852  poimirlem30  33071  hausgraph  37271
 Copyright terms: Public domain W3C validator