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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 2736 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | iscnrm 22174 | . 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
3 | 2 | simplbi 501 | 1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ∀wral 3051 𝒫 cpw 4499 ∪ cuni 4805 (class class class)co 7191 ↾t crest 16879 Topctop 21744 Nrmcnrm 22161 CNrmccnrm 22162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-cnrm 22169 |
This theorem is referenced by: restcnrm 22213 |
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