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Mirrors > Home > MPE Home > Th. List > nrmtop | Structured version Visualization version GIF version |
Description: A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
nrmtop | ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnrm 23161 | . 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ∩ cin 3939 ⊆ wss 3940 𝒫 cpw 4594 ‘cfv 6533 Topctop 22717 Clsdccld 22842 clsccl 22844 Nrmcnrm 23136 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-nrm 23143 |
This theorem is referenced by: pnrmtop 23167 nrmsep 23183 isnrm2 23184 isnrm3 23185 nrmr0reg 23575 kqnrm 23578 nrmhmph 23620 |
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