MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t1top Structured version   Visualization version   GIF version

Theorem t1top 21866
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 2818 . . 3 𝐽 = 𝐽
21ist1 21857 . 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 2105  wral 3135  {csn 4557   cuni 4830  cfv 6348  Topctop 21429  Clsdccld 21552  Frect1 21843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-t1 21850
This theorem is referenced by:  t1t0  21884  lpcls  21900  perfcls  21901  restt1  21903  t1sep2  21905  sst1  21910  t1connperf  21972  t1hmph  22327  qtopt1  30998  onint1  33694
  Copyright terms: Public domain W3C validator