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Theorem mstps 24348
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 24347 . 2 (𝑀 ∈ MetSp β†’ 𝑀 ∈ ∞MetSp)
2 xmstps 24346 . 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 2099  TopSpctps 22821  βˆžMetSpcxms 24210  MetSpcms 24211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-res 5684  df-iota 6494  df-fv 6550  df-xms 24213  df-ms 24214
This theorem is referenced by:  ngptps  24498  ngptgp  24532  cnfldtps  24681  cnmpt1ds  24745  cnmpt2ds  24746  rlmbn  25276  rrhcn  33534  sitgclbn  33899
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