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Theorem mstps 14627
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 14626 . 2  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
2 xmstps 14625 . 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 2164   TopSpctps 14198   *MetSpcxms 14504   MetSpcms 14505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-res 4671  df-iota 5215  df-fv 5262  df-xms 14507  df-ms 14508
This theorem is referenced by: (None)
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