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Theorem eldifpr 3978
Description: Membership in a set with two elements removed. Similar to eldifsn 4084 and eldiftp 4003. (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 3977 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) ) )
21notbid 294 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  -.  ( A  =  C  \/  A  =  D )
) )
3 neanior 2770 . . . 4  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  -.  ( A  =  C  \/  A  =  D )
)
42, 3syl6bbr 263 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  ( A  =/=  C  /\  A  =/= 
D ) ) )
54pm5.32i 637 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
6 eldif 3422 . 2  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D } ) )
7 3anass 969 . 2  |-  ( ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
85, 6, 73bitr4i 277 1  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  A  =/= 
C  /\  A  =/=  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641    \ cdif 3409   {cpr 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-v 3056  df-dif 3415  df-un 3417  df-sn 3962  df-pr 3964
This theorem is referenced by:  logbcl  26576  logbid1  26577  rnlogbval  26579  relogbcl  26581  logb1  26582  nnlogbexp  26583  rexdifpr  30248  elogb  31392
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