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Theorem t1top 22833
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.
StepHypRef Expression
1 eqid 2732 . . 3 𝐽 = 𝐽
21ist1 22824 . 2 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32simplbi 498 1 (𝐽 ∈ Fre → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3061  {csn 4628   cuni 4908  cfv 6543  Topctop 22394  Clsdccld 22519  Frect1 22810
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-t1 22817
This theorem is referenced by:  t1t0  22851  lpcls  22867  perfcls  22868  restt1  22870  t1sep2  22872  sst1  22877  t1connperf  22939  t1hmph  23294  qtopt1  32810  onint1  35329
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