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Theorem regtop 20280
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 20279 . 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 461 1  |-  ( J  e.  Reg  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   ` cfv 5601   Topctop 19848   clsccl 19964   Regcreg 20256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-reg 20263
This theorem is referenced by:  regsep2  20323  regr1  20696  kqreg  20697  reghmph  20739
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