MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stctop Structured version   Unicode version

Theorem 1stctop 17508
Description: A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Assertion
Ref Expression
1stctop  |-  ( J  e.  1stc  ->  J  e. 
Top )

Proof of Theorem 1stctop
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  U. J  =  U. J
21is1stc 17506 . 2  |-  ( J  e.  1stc  <->  ( J  e. 
Top  /\  A. x  e.  U. J E. y  e.  ~P  J ( y  ~<_  om  /\  A. z  e.  J  ( x  e.  z  ->  x  e. 
U. ( y  i^i 
~P z ) ) ) ) )
32simplbi 448 1  |-  ( J  e.  1stc  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321   ~Pcpw 3801   U.cuni 4017   class class class wbr 4214   omcom 4847    ~<_ cdom 7109   Topctop 16960   1stcc1stc 17502
This theorem is referenced by:  1stcfb  17510  1stcrest  17518  1stcelcls  17526  lly1stc  17561  1stckgen  17588  tx1stc  17684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-uni 4018  df-1stc 17504
  Copyright terms: Public domain W3C validator