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Theorem opprc1 4183
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4182. (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 457 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  A  e.  _V )
21con3i 135 . 2  |-  ( -.  A  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 4182 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 16 1  |-  ( -.  A  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   (/)c0 3738   <.cop 3984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-op 3985
This theorem is referenced by:  brprcneu  5785  eu2ndop1stv  30167  bj-inftyexpidisj  32842
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