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Theorem mstps 12658
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
mstps (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)

Proof of Theorem mstps
StepHypRef Expression
1 msxms 12657 . 2 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
2 xmstps 12656 . 2 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
31, 2syl 14 1 (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  TopSpctps 12227  ∞MetSpcxms 12535  MetSpcms 12536
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-rab 2426  df-v 2689  df-un 3076  df-in 3078  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-xp 4549  df-res 4555  df-iota 5092  df-fv 5135  df-xms 12538  df-ms 12539
This theorem is referenced by: (None)
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