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Theorem nrmtop 23191
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 23190 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
21simplbi 497 1 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3055  wrex 3064  cin 3942  wss 3943  𝒫 cpw 4597  cfv 6536  Topctop 22746  Clsdccld 22871  clsccl 22873  Nrmcnrm 23165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-nrm 23172
This theorem is referenced by:  pnrmtop  23196  nrmsep  23212  isnrm2  23213  isnrm3  23214  nrmr0reg  23604  kqnrm  23607  nrmhmph  23649
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