MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mstps Structured version   Unicode version

Theorem mstps 20785
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 20784 . 2  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
2 xmstps 20783 . 2  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
31, 2syl 16 1  |-  ( M  e.  MetSp  ->  M  e.  TopSp
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   TopSpctps 19204   *MetSpcxme 20647   MetSpcmt 20648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5551  df-fv 5596  df-xms 20650  df-ms 20651
This theorem is referenced by:  ngptps  20949  ngptgp  20977  cnfldtps  21112  cnmpt1ds  21174  cnmpt2ds  21175  rlmbn  21628  rrhcn  27729  sitgclbn  28036
  Copyright terms: Public domain W3C validator