Theorem t1top 23675

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 2763	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist1 23666	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080  {csn 4588  ∪ cuni 4870  ‘cfv 6555  Topctop 23238  Clsdccld 23361  Frect1 23652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-t1 23659

This theorem is used by:  t1t0  23693  lpcls  23709  perfcls  23710  restt1  23712  t1sep2  23714  sst1  23719  t1connperf  23781  t1hmph  24137  qtopt1  34534  onint1  37414