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Theorem oprcl 3800
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 2753 . 2  |-  ( C  e.  <. A ,  B >.  ->  E. y  y  e. 
<. A ,  B >. )
2 df-op 3600 . . . . . . 7  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
32eleq2i 2244 . . . . . 6  |-  ( y  e.  <. A ,  B >.  <-> 
y  e.  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) } )
4 df-clab 2164 . . . . . 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 994 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76sbimi 1764 . . . . 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 1528 . . . . 5  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
109sbf 1777 . . . 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 1598 . 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 978   E.wex 1492   [wsb 1762    e. wcel 2148   {cab 2163   _Vcvv 2737   {csn 3591   {cpr 3592   <.cop 3594
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739  df-op 3600
This theorem is referenced by:  opth1  4232  opth  4233  0nelop  4244
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