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Theorem mstps 24486
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 24485 . 2 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
2 xmstps 24484 . 2 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
31, 2syl 17 1 (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  TopSpctps 22959  ∞MetSpcxms 24348  MetSpcms 24349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712  df-iota 6525  df-fv 6581  df-xms 24351  df-ms 24352
This theorem is referenced by:  ngptps  24636  ngptgp  24670  cnfldtps  24819  cnmpt1ds  24883  cnmpt2ds  24884  rlmbn  25414  rrhcn  33943  sitgclbn  34308
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