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Theorem oprcl 3641
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
oprcl  |-  ( C  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )

Proof of Theorem oprcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex2 2635 . 2  |-  ( C  e.  <. A ,  B >.  ->  E. y  y  e. 
<. A ,  B >. )
2 df-op 3450 . . . . . . 7  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
32eleq2i 2154 . . . . . 6  |-  ( y  e.  <. A ,  B >.  <-> 
y  e.  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) } )
4 df-clab 2075 . . . . . 6  |-  ( y  e.  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  <->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
53, 4bitri 182 . . . . 5  |-  ( y  e.  <. A ,  B >.  <->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
6 3simpa 940 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76sbimi 1694 . . . . 5  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
85, 7sylbi 119 . . . 4  |-  ( y  e.  <. A ,  B >.  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
9 nfv 1466 . . . . 5  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
109sbf 1707 . . . 4  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V )  <->  ( A  e.  _V  /\  B  e.  _V )
)
118, 10sylib 120 . . 3  |-  ( y  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1211exlimiv 1534 . 2  |-  ( E. y  y  e.  <. A ,  B >.  ->  ( A  e.  _V  /\  B  e.  _V ) )
131, 12syl 14 1  |-  ( C  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924   E.wex 1426    e. wcel 1438   [wsb 1692   {cab 2074   _Vcvv 2619   {csn 3441   {cpr 3442   <.cop 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-op 3450
This theorem is referenced by:  opth1  4054  opth  4055  0nelop  4066
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