Theorem regtop 23357

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 23356	. 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 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ‘cfv 6563  Topctop 22915  clsccl 23042  Regcreg 23333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-reg 23340

This theorem is referenced by:  regsep2  23400  regr1  23774  kqreg  23775  reghmph  23817