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

Theorem tpsexOLD 16657
Description: Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tpsexOLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )

Proof of Theorem tpsexOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4024 . 2  |-  ( A
TopSp OLD J  <->  <. A ,  J >.  e.  TopSp OLD )
2 df-topspOLD 16637 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
32relopabi 4811 . . 3  |-  Rel  TopSp OLD
4 brrelex12 4726 . . 3  |-  ( ( Rel  TopSp OLD  /\  A TopSp OLD J )  ->  ( A  e.  _V  /\  J  e.  _V ) )
53, 4mpan 651 . 2  |-  ( A
TopSp OLD J  ->  ( A  e.  _V  /\  J  e.  _V ) )
61, 5sylbir 204 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   U.cuni 3827   class class class wbr 4023   Rel wrel 4694   Topctop 16631   TopSp OLDctpsOLD 16633
This theorem is referenced by:  istpsOLD  16658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-topspOLD 16637
  Copyright terms: Public domain W3C validator