Theorem haustop 20945

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 2610	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus 20936	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
3	2	simplbi 475	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  ∅c0 3874  ∪ cuni 4372  Topctop 20517  Hauscha 20922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-uni 4373  df-haus 20929

This theorem is referenced by:  haust1  20966  resthaus  20982  sshaus  20989  lmmo  20994  hauscmplem  21019  hauscmp  21020  hauslly  21105  hausllycmp  21107  kgenhaus  21157  pthaus  21251  txhaus  21260  xkohaus  21266  haushmph  21405  cmphaushmeo  21413  hausflim  21595  hauspwpwf1  21601  hauspwpwdom  21602  hausflf  21611  cnextfun  21678  cnextfvval  21679  cnextf  21680  cnextcn  21681  cnextfres1  21682  cnextfres  21683  qtophaus  29231  ismntop  29398  poimirlem30  32609  hausgraph  36809