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Theorem mstps 18446
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
mstps  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)

Proof of Theorem mstps
StepHypRef Expression
1 msxms 18445 . 2  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )
2 xmstps 18444 . 2  |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
31, 2syl 16 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   TopSpctps 16924   * MetSpcxme 18308   MetSpcmt 18309
This theorem is referenced by:  ngptps  18610  ngptgp  18638  cnfldtps  18773  cnmpt1ds  18834  cnmpt2ds  18835  rlmbn  19276  rrhre  24348  sitgclbn  24618
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-res 4857  df-iota 5385  df-fv 5429  df-xms 18311  df-ms 18312
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