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Theorem prprc 3547
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 3545 . 2  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
2 snprc 3502 . . 3  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 118 . 2  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
41, 3sylan9eq 2140 1  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   (/)c0 3284   {csn 3441   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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