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Theorem eldiftp 23395
Description: Membership in a set with three elements removed. Similar to eldifsn 3749 and eldifpr 23394. (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 3162 . 2  |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D ,  E } ) )
2 eltpg 3676 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D ,  E }  <->  ( A  =  C  \/  A  =  D  \/  A  =  E )
) )
32notbid 285 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D ,  E }  <->  -.  ( A  =  C  \/  A  =  D  \/  A  =  E ) ) )
4 ne3anior 2532 . . . 4  |-  ( ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E )  <->  -.  ( A  =  C  \/  A  =  D  \/  A  =  E )
)
53, 4syl6bbr 254 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D ,  E }  <->  ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E ) ) )
65pm5.32i 618 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D ,  E }
)  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/= 
E ) ) )
71, 6bitri 240 1  |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/= 
E ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {ctp 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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