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Theorem opprc2 3572
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3570. (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 103 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  B  e.  _V )
21con3i 562 . 2  |-  ( -.  B  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 3570 . 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 97    = wceq 1243    e. wcel 1393   _Vcvv 2557   (/)c0 3224   <.cop 3378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-op 3384
This theorem is referenced by: (None)
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