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Theorem eldiftp 4046
Description: Membership in a set with three elements removed. Similar to eldifsn 4128 and eldifpr 4019. (Contributed by David A. Wheeler, 22-Jul-2017.)
Assertion
Ref Expression
eldiftp  |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/= 
E ) ) )

Proof of Theorem eldiftp
StepHypRef Expression
1 eldif 3452 . 2  |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D ,  E } ) )
2 eltpg 4045 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D ,  E }  <->  ( A  =  C  \/  A  =  D  \/  A  =  E )
) )
32notbid 295 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D ,  E }  <->  -.  ( A  =  C  \/  A  =  D  \/  A  =  E ) ) )
4 ne3anior 2757 . . . 4  |-  ( ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E )  <->  -.  ( A  =  C  \/  A  =  D  \/  A  =  E )
)
53, 4syl6bbr 266 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D ,  E }  <->  ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E ) ) )
65pm5.32i 641 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D ,  E }
)  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/= 
E ) ) )
71, 6bitri 252 1  |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/= 
E ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439   {ctp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-sn 4003  df-pr 4005  df-tp 4007
This theorem is referenced by: (None)
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