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Theorem tpsexOLD 16673
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 4040 . 2  |-  ( A
TopSp OLD J  <->  <. A ,  J >.  e.  TopSp OLD )
2 df-topspOLD 16653 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
32relopabi 4827 . . 3  |-  Rel  TopSp OLD
4 brrelex12 4742 . . 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   U.cuni 3843   class class class wbr 4039   Rel wrel 4710   Topctop 16647   TopSp OLDctpsOLD 16649
This theorem is referenced by:  istpsOLD  16674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-topspOLD 16653
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