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Theorem prprc 3751
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 3749 . 2  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
2 snprc 3708 . . 3  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 186 . 2  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
41, 3sylan9eq 2348 1  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {csn 3653   {cpr 3654
This theorem is referenced by:  usgraedgprv  28256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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