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Theorem mstps 23961
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
mstps (𝑀 ∈ MetSp β†’ 𝑀 ∈ TopSp)

Proof of Theorem mstps
StepHypRef Expression
1 msxms 23960 . 2 (𝑀 ∈ MetSp β†’ 𝑀 ∈ ∞MetSp)
2 xmstps 23959 . 2 (𝑀 ∈ ∞MetSp β†’ 𝑀 ∈ TopSp)
31, 2syl 17 1 (𝑀 ∈ MetSp β†’ 𝑀 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  TopSpctps 22434  βˆžMetSpcxms 23823  MetSpcms 23824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-iota 6496  df-fv 6552  df-xms 23826  df-ms 23827
This theorem is referenced by:  ngptps  24111  ngptgp  24145  cnfldtps  24294  cnmpt1ds  24358  cnmpt2ds  24359  rlmbn  24878  rrhcn  32977  sitgclbn  33342
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