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Theorem opprc 3696
Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )

Proof of Theorem opprc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-op 3506 . 2  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
2 3simpa 963 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32con3i 606 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
43alrimiv 1830 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  A. x  -.  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } ) )
5 abeq0 3363 . . 3  |-  ( { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } ) }  =  (/)  <->  A. x  -.  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } ) )
64, 5sylibr 133 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
71, 6syl5eq 2162 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    /\ w3a 947   A.wal 1314    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660   (/)c0 3333   {csn 3497   {cpr 3498   <.cop 3500
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334  df-op 3506
This theorem is referenced by:  opprc1  3697  opprc2  3698  ovprc  5774
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