Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnrmtop Structured version   Visualization version   GIF version

Theorem pnrmtop 21366
 Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
pnrmtop (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop
StepHypRef Expression
1 pnrmnrm 21365 . 2 (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2 nrmtop 21361 . 2 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
31, 2syl 17 1 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145  Topctop 20918  Nrmcnrm 21335  PNrmcpnrm 21337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-cnv 5258  df-dm 5260  df-rn 5261  df-iota 5993  df-fv 6038  df-ov 6799  df-nrm 21342  df-pnrm 21344 This theorem is referenced by:  pnrmopn  21368
 Copyright terms: Public domain W3C validator