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Theorem t1top 21355
 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 2771 . . 3 𝐽 = 𝐽
21ist1 21346 . 2 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32simplbi 485 1 (𝐽 ∈ Fre → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145  ∀wral 3061  {csn 4316  ∪ cuni 4574  ‘cfv 6031  Topctop 20918  Clsdccld 21041  Frect1 21332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-t1 21339 This theorem is referenced by:  t1t0  21373  lpcls  21389  perfcls  21390  restt1  21392  t1sep2  21394  sst1  21399  t1connperf  21460  t1hmph  21815  qtopt1  30242  onint1  32785
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