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Theorem opprc2 3622
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3620. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc2  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )

Proof of Theorem opprc2
StepHypRef Expression
1 simpr 108 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  B  e.  _V )
21con3i 595 . 2  |-  ( -.  B  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 3620 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 14 1  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   _Vcvv 2614   (/)c0 3272   <.cop 3428
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 2616  df-dif 2988  df-nul 3273  df-op 3434
This theorem is referenced by: (None)
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