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Theorem oprcl 3736
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 2705 . 2  |-  ( C  e.  <. A ,  B >.  ->  E. y  y  e. 
<. A ,  B >. )
2 df-op 3540 . . . . . . 7  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
32eleq2i 2207 . . . . . 6  |-  ( y  e.  <. A ,  B >.  <-> 
y  e.  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) } )
4 df-clab 2127 . . . . . 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 183 . . . . 5  |-  ( y  e.  <. A ,  B >.  <->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
6 3simpa 979 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76sbimi 1738 . . . . 5  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
85, 7sylbi 120 . . . 4  |-  ( y  e.  <. A ,  B >.  ->  [ y  /  x ] ( A  e. 
_V  /\  B  e.  _V ) )
9 nfv 1509 . . . . 5  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
109sbf 1751 . . . 4  |-  ( [ y  /  x ]
( A  e.  _V  /\  B  e.  _V )  <->  ( A  e.  _V  /\  B  e.  _V )
)
118, 10sylib 121 . . 3  |-  ( y  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1211exlimiv 1578 . 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 103    /\ w3a 963   E.wex 1469    e. wcel 1481   [wsb 1736   {cab 2126   _Vcvv 2689   {csn 3531   {cpr 3532   <.cop 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-op 3540
This theorem is referenced by:  opth1  4165  opth  4166  0nelop  4177
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