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Theorem prprc 3916
Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )

Proof of Theorem prprc
StepHypRef Expression
1 prprc1 3914 . 2  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
2 snprc 3871 . . 3  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 187 . 2  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
41, 3sylan9eq 2488 1  |-  ( ( -.  A  e.  _V  /\ 
-.  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 2956   (/)c0 3628   {csn 3814   {cpr 3815
This theorem is referenced by:  usgraedgprv  21396
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 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-sn 3820  df-pr 3821
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