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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 23363 | . 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3963 ∩ cint 4951 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℕcn 12264 Clsdccld 23040 Nrmcnrm 23334 PNrmcpnrm 23336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 df-iota 6516 df-fv 6571 df-ov 7434 df-pnrm 23343 |
This theorem is referenced by: pnrmtop 23365 |
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