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Theorem mstps 12901
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 12900 . 2  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
2 xmstps 12899 . 2  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
31, 2syl 14 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   TopSpctps 12470   *MetSpcxms 12778   MetSpcms 12779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-xp 4593  df-res 4599  df-iota 5136  df-fv 5179  df-xms 12781  df-ms 12782
This theorem is referenced by: (None)
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