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Theorem opprc2 4155
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4153. (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 459 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  B  e.  _V )
21con3i 135 . 2  |-  ( -.  B  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 opprc 4153 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
42, 3syl 16 1  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   (/)c0 3711   <.cop 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-op 3951
This theorem is referenced by:  dmsnopss  5388  strle1  14733
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