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Theorem prprc 3597
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 3595 . 2  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
2 snprc 3552 . . 3  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 119 . 2  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
41, 3sylan9eq 2165 1  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1312    e. wcel 1461   _Vcvv 2655   (/)c0 3327   {csn 3491   {cpr 3492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-un 3039  df-nul 3328  df-sn 3497  df-pr 3498
This theorem is referenced by: (None)
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