Theorem t1top 21134

Description: A T₁ space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
t1top	|- ( J e. Fre -> J e. Top )

Proof of Theorem t1top

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	ist1 21125	. 2 |- ( J e. Fre <-> ( J e. Top /\ A. x e. U. J { x } e. ( Clsd ` J ) ) )
3	2	simplbi 476	1 |- ( J e. Fre -> J e. Top )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820   Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-t1 21118

This theorem is referenced by:  t1t0  21152  lpcls  21168  perfcls  21169  restt1  21171  t1sep2  21173  sst1  21178  t1connperf  21239  t1hmph  21594  qtopt1  29902  onint1  32448