Theorem t1top 23295

Description: A T₁ space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
t1top	⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)

Proof of Theorem t1top

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist1 23286	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3051  {csn 4567  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  Clsdccld 22981  Frect1 23272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-t1 23279

This theorem is referenced by:  t1t0  23313  lpcls  23329  perfcls  23330  restt1  23332  t1sep2  23334  sst1  23339  t1connperf  23401  t1hmph  23756  qtopt1  33979  onint1  36631