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Theorem cnrmtop 17325
Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmtop  |-  ( J  e. CNrm  ->  J  e.  Top )

Proof of Theorem cnrmtop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . . 3  |-  U. J  =  U. J
21iscnrm 17311 . 2  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
32simplbi 447 1  |-  ( J  e. CNrm  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   A.wral 2651   ~Pcpw 3744   U.cuni 3959  (class class class)co 6022   ↾t crest 13577   Topctop 16883   Nrmcnrm 17298  CNrmccnrm 17299
This theorem is referenced by:  restcnrm  17350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-ov 6025  df-cnrm 17306
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