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Theorem nrmtop 17431
Description: A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
nrmtop  |-  ( J  e.  Nrm  ->  J  e.  Top )

Proof of Theorem nrmtop
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnrm 17430 . 2  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
21simplbi 448 1  |-  ( J  e.  Nrm  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   A.wral 2711   E.wrex 2712    i^i cin 3305    C_ wss 3306   ~Pcpw 3823   ` cfv 5483   Topctop 16989   Clsdccld 17111   clsccl 17113   Nrmcnrm 17405
This theorem is referenced by:  pnrmtop  17436  nrmsep  17452  isnrm2  17453  isnrm3  17454  nrmr0reg  17812  kqnrm  17815  nrmhmph  17857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-nrm 17412
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