Theorem t1top 22044

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 2758	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist1 22035	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3070  {csn 4525  ∪ cuni 4801  ‘cfv 6340  Topctop 21607  Clsdccld 21730  Frect1 22021

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-t1 22028

This theorem is referenced by:  t1t0  22062  lpcls  22078  perfcls  22079  restt1  22081  t1sep2  22083  sst1  22088  t1connperf  22150  t1hmph  22505  qtopt1  31319  onint1  34222