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Mirrors > Home > MPE Home > Th. List > cnrmtop | Structured version Visualization version GIF version |
Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
cnrmtop | ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | iscnrm 22819 | . 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
3 | 2 | simplbi 499 | 1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3062 𝒫 cpw 4602 ∪ cuni 4908 (class class class)co 7406 ↾t crest 17363 Topctop 22387 Nrmcnrm 22806 CNrmccnrm 22807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6493 df-fv 6549 df-ov 7409 df-cnrm 22814 |
This theorem is referenced by: restcnrm 22858 |
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