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Theorem nelpri 4048
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
nelpri.1  |-  A  =/= 
B
nelpri.2  |-  A  =/= 
C
Assertion
Ref Expression
nelpri  |-  -.  A  e.  { B ,  C }

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2  |-  A  =/= 
B
2 nelpri.2 . 2  |-  A  =/= 
C
3 neanior 2792 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 4047 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 135 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 195 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6mp2an 672 1  |-  -.  A  e.  { B ,  C }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  constr3pthlem1  24331  konigsberg  24663  ex-dif  24821  ex-in  24823  ex-pss  24826  ex-res  24839  AnelBC  32239
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