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Theorem mstps 18103
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 18102 . 2  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )
2 xmstps 18101 . 2  |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
31, 2syl 15 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710   TopSpctps 16740   * MetSpcxme 17984   MetSpcmt 17985
This theorem is referenced by:  ngptps  18226  ngptgp  18254  cnfldtps  18389  cnmpt1ds  18450  cnmpt2ds  18451  rlmbn  18882
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-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-opab 4159  df-xp 4777  df-res 4783  df-iota 5301  df-fv 5345  df-xms 17987  df-ms 17988
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