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Theorem opprc2 3999
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3997. (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 448 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  B  e.  _V )
21con3i 129 . 2  |-  ( -.  B  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 3997 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 16 1  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   <.cop 3809
This theorem is referenced by:  dmsnopss  5334  strle1  13552
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-op 3815
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