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Theorem tpsexOLD 16984
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 4213 . 2  |-  ( A
TopSp OLD J  <->  <. A ,  J >.  e.  TopSp OLD )
2 df-topspOLD 16964 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
32relopabi 5000 . . 3  |-  Rel  TopSp OLD
4 brrelex12 4915 . . 3  |-  ( ( Rel  TopSp OLD  /\  A TopSp OLD J )  ->  ( A  e.  _V  /\  J  e.  _V ) )
53, 4mpan 652 . 2  |-  ( A
TopSp OLD J  ->  ( A  e.  _V  /\  J  e.  _V ) )
61, 5sylbir 205 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   U.cuni 4015   class class class wbr 4212   Rel wrel 4883   Topctop 16958   TopSp OLDctpsOLD 16960
This theorem is referenced by:  istpsOLD  16985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-topspOLD 16964
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