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Theorem eldifpr 3587
Description: Membership in a set with two elements removed. Similar to eldifsn 3687 and eldiftp 3606. (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 3580 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) ) )
21notbid 657 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  -.  ( A  =  C  \/  A  =  D )
) )
3 neanior 2414 . . . 4  |-  ( ( A  =/=  C  /\  A  =/=  D )  <->  -.  ( A  =  C  \/  A  =  D )
)
42, 3bitr4di 197 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  ( A  =/=  C  /\  A  =/= 
D ) ) )
54pm5.32i 450 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
6 eldif 3111 . 2  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D } ) )
7 3anass 967 . 2  |-  ( ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
85, 6, 73bitr4i 211 1  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  A  =/= 
C  /\  A  =/=  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1335    e. wcel 2128    =/= wne 2327    \ cdif 3099   {cpr 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-un 3106  df-sn 3566  df-pr 3567
This theorem is referenced by:  rexdifpr  3588  rplogbval  13304
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