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Theorem opprc1 4226
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4225. (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 455 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  A  e.  _V )
21con3i 135 . 2  |-  ( -.  A  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 4225 . 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   <.cop 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-op 4023
This theorem is referenced by:  brprcneu  5841  eu2ndop1stv  32446  bj-inftyexpidisj  35013
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