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Theorem eldifpr 24181
Description: Membership in a set with two elements removed. Similar to eldifsn 3863 and eldiftp 24182. (Contributed by Mario Carneiro, 18-Jul-2017.)
Assertion
Ref Expression
eldifpr  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  A  =/= 
C  /\  A  =/=  D ) )

Proof of Theorem eldifpr
StepHypRef Expression
1 elprg 3767 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) ) )
21notbid 286 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  -.  ( A  =  C  \/  A  =  D )
) )
3 neanior 2628 . . . 4  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  -.  ( A  =  C  \/  A  =  D )
)
42, 3syl6bbr 255 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  ( A  =/=  C  /\  A  =/= 
D ) ) )
54pm5.32i 619 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
6 eldif 3266 . 2  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D } ) )
7 3anass 940 . 2  |-  ( ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
85, 6, 73bitr4i 269 1  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  A  =/= 
C  /\  A  =/=  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253   {cpr 3751
This theorem is referenced by:  logbcl  24186  logbid1  24187  rnlogbval  24189  relogbcl  24191  logb1  24192  nnlogbexp  24193  elogb  27871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-un 3261  df-sn 3756  df-pr 3757
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