Theorem nrmtop 23326

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 23325	. 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 396   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536  ‘cfv 6492  Topctop 22883  Clsdccld 23006  clsccl 23008  Nrmcnrm 23300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-nrm 23307

This theorem is referenced by:  pnrmtop  23331  nrmsep  23347  isnrm2  23348  isnrm3  23349  nrmr0reg  23739  kqnrm  23742  nrmhmph  23784