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Theorem opprc 3626
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 3440 . 2  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
2 3simpa 938 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32con3i 595 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) )
43alrimiv 1799 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  A. x  -.  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } ) )
5 abeq0 3302 . . 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 132 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
71, 6syl5eq 2129 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 922   A.wal 1285    = wceq 1287    e. wcel 1436   {cab 2071   _Vcvv 2615   (/)c0 3275   {csn 3431   {cpr 3432   <.cop 3434
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-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-nul 3276  df-op 3440
This theorem is referenced by:  opprc1  3627  opprc2  3628  ovprc  5641
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