Theorem nrmtop 23162

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.

Step	Hyp	Ref	Expression
1		isnrm 23161	. 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
2	1	simplbi 497	1 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062   ∩ cin 3939   ⊆ wss 3940  𝒫 cpw 4594  ‘cfv 6533  Topctop 22717  Clsdccld 22842  clsccl 22844  Nrmcnrm 23136

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 2695

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 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-nrm 23143

This theorem is referenced by:  pnrmtop  23167  nrmsep  23183  isnrm2  23184  isnrm3  23185  nrmr0reg  23575  kqnrm  23578  nrmhmph  23620