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Theorem regtop 19628
Description: A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
regtop  |-  ( J  e.  Reg  ->  J  e.  Top )

Proof of Theorem regtop
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isreg 19627 . 2  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  (
( cls `  J
) `  z )  C_  x ) ) )
21simplbi 460 1  |-  ( J  e.  Reg  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ` cfv 5588   Topctop 19189   clsccl 19313   Regcreg 19604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-reg 19611
This theorem is referenced by:  regsep2  19671  regr1  20014  kqreg  20015  reghmph  20057
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