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Theorem opprc1 4006
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4005. (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 444 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  A  e.  _V )
21con3i 129 . 2  |-  ( -.  A  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 4005 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 16 1  |-  ( -.  A  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   <.cop 3817
This theorem is referenced by:  brprcneu  5721  eu2ndop1stv  27956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-op 3823
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