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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 22767 | . 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 3941 ∩ cint 4940 ↦ cmpt 5221 ran crn 5667 ‘cfv 6529 (class class class)co 7390 ↑m cmap 8800 ℕcn 12191 Clsdccld 22444 Nrmcnrm 22738 PNrmcpnrm 22740 |
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 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-cnv 5674 df-dm 5676 df-rn 5677 df-iota 6481 df-fv 6537 df-ov 7393 df-pnrm 22747 |
This theorem is referenced by: pnrmtop 22769 |
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