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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 22486 | . 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 ‘cfv 6433 Topctop 22042 Clsdccld 22167 clsccl 22169 Nrmcnrm 22461 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-nrm 22468 |
This theorem is referenced by: pnrmtop 22492 nrmsep 22508 isnrm2 22509 isnrm3 22510 nrmr0reg 22900 kqnrm 22903 nrmhmph 22945 |
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