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Theorem opprc1 3818
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 3817. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc1  |-  ( -.  A  e.  _V  ->  <. A ,  B >.  =  (/) )

Proof of Theorem opprc1
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  A  e.  _V )
21con3i 127 . 2  |-  ( -.  A  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 3817 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 15 1  |-  ( -.  A  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643
This theorem is referenced by:  brprcneu  5518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-op 3649
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