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Theorem t1top 22815
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 2733 . . 3 𝐽 = 𝐽
21ist1 22806 . 2 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32simplbi 499 1 (𝐽 ∈ Fre → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3062  {csn 4626   cuni 4906  cfv 6539  Topctop 22376  Clsdccld 22501  Frect1 22792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-iota 6491  df-fv 6547  df-t1 22799
This theorem is referenced by:  t1t0  22833  lpcls  22849  perfcls  22850  restt1  22852  t1sep2  22854  sst1  22859  t1connperf  22921  t1hmph  23276  qtopt1  32752  onint1  35271
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