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Theorem mstps 20030
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 20029 . 2  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
2 xmstps 20028 . 2  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
31, 2syl 16 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   TopSpctps 18501   *MetSpcxme 19892   MetSpcmt 19893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-xp 4846  df-res 4852  df-iota 5381  df-fv 5426  df-xms 19895  df-ms 19896
This theorem is referenced by:  ngptps  20194  ngptgp  20222  cnfldtps  20357  cnmpt1ds  20419  cnmpt2ds  20420  rlmbn  20873  rrhcn  26426  sitgclbn  26729
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