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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 24404 | . 2 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) | |
2 | xmstps 24403 | . 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 2098 TopSpctps 22878 ∞MetSpcxms 24267 MetSpcms 24268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-res 5690 df-iota 6501 df-fv 6557 df-xms 24270 df-ms 24271 |
This theorem is referenced by: ngptps 24555 ngptgp 24589 cnfldtps 24738 cnmpt1ds 24802 cnmpt2ds 24803 rlmbn 25333 rrhcn 33729 sitgclbn 34094 |
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