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Theorem nrmtop 17170
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 17169 . 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 446 1  |-  ( J  e.  Nrm  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620    i^i cin 3227    C_ wss 3228   ~Pcpw 3701   ` cfv 5337   Topctop 16737   Clsdccld 16859   clsccl 16861   Nrmcnrm 17144
This theorem is referenced by:  pnrmtop  17175  nrmsep  17191  isnrm2  17192  isnrm3  17193  nrmr0reg  17546  kqnrm  17549  nrmhmph  17591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-nrm 17151
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