Theorem regtop 23276

Description: A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
regtop	⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)

Proof of Theorem regtop

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isreg 23275	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
2	1	simplbi 497	1 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931  ‘cfv 6536  Topctop 22836  clsccl 22961  Regcreg 23252

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-reg 23259

This theorem is referenced by:  regsep2  23319  regr1  23693  kqreg  23694  reghmph  23736