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Theorem mstps 21401
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 21400 . 2  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
2 xmstps 21399 . 2  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
31, 2syl 17 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   TopSpctps 19850   *MetSpcxme 21263   MetSpcmt 21264
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-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-opab 4485  df-xp 4860  df-res 4866  df-iota 5565  df-fv 5609  df-xms 21266  df-ms 21267
This theorem is referenced by:  ngptps  21547  ngptgp  21575  cnfldtps  21709  cnmpt1ds  21771  cnmpt2ds  21772  rlmbn  22221  rrhcn  28640  sitgclbn  29004
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