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Mirrors > Home > MPE Home > Th. List > pnrmnrm | Structured version Visualization version GIF version |
Description: A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
pnrmnrm | ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispnrm 22727 | . 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3913 ∩ cint 4912 ↦ cmpt 5193 ran crn 5639 ‘cfv 6501 (class class class)co 7362 ↑m cmap 8772 ℕcn 12162 Clsdccld 22404 Nrmcnrm 22698 PNrmcpnrm 22700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3406 df-v 3448 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-cnv 5646 df-dm 5648 df-rn 5649 df-iota 6453 df-fv 6509 df-ov 7365 df-pnrm 22707 |
This theorem is referenced by: pnrmtop 22729 |
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