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Theorem opprc2 3951
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3949. (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 3949 . 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 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573   <.cop 3762
This theorem is referenced by:  dmsnopss  5284  strle1  13489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-op 3768
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