Theorem mstps 24341

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

Step	Hyp	Ref	Expression
1		msxms 24340	. 2 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
2		xmstps 24339	. 2 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2109  TopSpctps 22817  ∞MetSpcxms 24203  MetSpcms 24204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-res 5631  df-iota 6438  df-fv 6490  df-xms 24206  df-ms 24207

This theorem is referenced by:  ngptps  24488  ngptgp  24522  cnfldtps  24663  cnmpt1ds  24729  cnmpt2ds  24730  rlmbn  25259  rrhcn  33970  sitgclbn  34317