Theorem t1top 23648

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 2760	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist1 23639	. 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 2144  ∀wral 3076  {csn 4583  ∪ cuni 4866  ‘cfv 6538  Topctop 23211  Clsdccld 23334  Frect1 23625

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-t1 23632

This theorem is used by:  t1t0  23666  lpcls  23682  perfcls  23683  restt1  23685  t1sep2  23687  sst1  23692  t1connperf  23754  t1hmph  24110  qtopt1  34467  onint1  37237