Theorem mstps 24349

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 24348	. 2 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
2		xmstps 24347	. 2 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2109  TopSpctps 22825  ∞MetSpcxms 24211  MetSpcms 24212

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-xp 5652  df-res 5658  df-iota 6472  df-fv 6527  df-xms 24214  df-ms 24215

This theorem is referenced by:  ngptps  24496  ngptgp  24530  cnfldtps  24671  cnmpt1ds  24737  cnmpt2ds  24738  rlmbn  25268  rrhcn  33995  sitgclbn  34342