Theorem t1top 23456

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 2769	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist1 23447	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085  {csn 4592  ∪ cuni 4874  ‘cfv 6537  Topctop 23019  Clsdccld 23142  Frect1 23433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-iota 6493  df-fv 6545  df-t1 23440

This theorem is referenced by:  t1t0  23474  lpcls  23490  perfcls  23491  restt1  23493  t1sep2  23495  sst1  23500  t1connperf  23562  t1hmph  23917  qtopt1  34170  onint1  36883