Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mstps | Structured version Visualization version GIF version |
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
mstps | ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msxms 23058 | . 2 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) | |
2 | xmstps 23057 | . 2 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 TopSpctps 21534 ∞MetSpcxms 22921 MetSpcms 22922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-res 5562 df-iota 6309 df-fv 6358 df-xms 22924 df-ms 22925 |
This theorem is referenced by: ngptps 23205 ngptgp 23239 cnfldtps 23380 cnmpt1ds 23444 cnmpt2ds 23445 rlmbn 23958 rrhcn 31233 sitgclbn 31596 |
Copyright terms: Public domain | W3C validator |