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Theorem regtop 21463
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.
StepHypRef Expression
1 isreg 21462 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
21simplbi 492 1 (𝐽 ∈ Reg → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3087  wrex 3088  wss 3767  cfv 6099  Topctop 21023  clsccl 21148  Regcreg 21439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107  df-reg 21446
This theorem is referenced by:  regsep2  21506  regr1  21879  kqreg  21880  reghmph  21922
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