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

Theorem nrmtop 21121
Description: A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
nrmtop (𝐽 ∈ Nrm → 𝐽 ∈ Top)

Proof of Theorem nrmtop
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnrm 21120 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
21simplbi 476 1 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  wral 2909  wrex 2910  cin 3566  wss 3567  𝒫 cpw 4149  cfv 5876  Topctop 20679  Clsdccld 20801  clsccl 20803  Nrmcnrm 21095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-nrm 21102
This theorem is referenced by:  pnrmtop  21126  nrmsep  21142  isnrm2  21143  isnrm3  21144  nrmr0reg  21533  kqnrm  21536  nrmhmph  21578
  Copyright terms: Public domain W3C validator