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Theorem oprcl 3820
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 2768 . 2  |-  ( C  e.  <. A ,  B >.  ->  E. y  y  e. 
<. A ,  B >. )
2 df-op 3619 . . . . . . 7  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
32eleq2i 2256 . . . . . 6  |-  ( y  e.  <. A ,  B >.  <-> 
y  e.  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) } )
4 df-clab 2176 . . . . . 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 184 . . . . 5  |-  ( y  e.  <. A ,  B >.  <->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
6 3simpa 996 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76sbimi 1775 . . . . 5  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
85, 7sylbi 121 . . . 4  |-  ( y  e.  <. A ,  B >.  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
9 nfv 1539 . . . . 5  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
109sbf 1788 . . . 4  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V )  <->  ( A  e.  _V  /\  B  e.  _V )
)
118, 10sylib 122 . . 3  |-  ( y  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1211exlimiv 1609 . 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 104    /\ w3a 980   E.wex 1503   [wsb 1773    e. wcel 2160   {cab 2175   _Vcvv 2752   {csn 3610   {cpr 3611   <.cop 3613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754  df-op 3619
This theorem is referenced by:  opth1  4257  opth  4258  0nelop  4269
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