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Theorem t1top 21115
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 2620 . . 3 𝐽 = 𝐽
21ist1 21106 . 2 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32simplbi 476 1 (𝐽 ∈ Fre → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wral 2909  {csn 4168   cuni 4427  cfv 5876  Topctop 20679  Clsdccld 20801  Frect1 21092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-t1 21099
This theorem is referenced by:  t1t0  21133  lpcls  21149  perfcls  21150  restt1  21152  t1sep2  21154  sst1  21159  t1connperf  21220  t1hmph  21575  qtopt1  29876  onint1  32423
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