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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 23343 | . 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 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ‘cfv 6561 Topctop 22899 Clsdccld 23024 clsccl 23026 Nrmcnrm 23318 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-nrm 23325 |
| This theorem is referenced by: pnrmtop 23349 nrmsep 23365 isnrm2 23366 isnrm3 23367 nrmr0reg 23757 kqnrm 23760 nrmhmph 23802 |
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