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Theorem eldifpr 3610
Description: Membership in a set with two elements removed. Similar to eldifsn 3710 and eldiftp 3629. (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 3603 . . . . 5  |-  ( A  e.  B  ->  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) ) )
21notbid 662 . . . 4  |-  ( A  e.  B  ->  ( -.  A  e.  { C ,  D }  <->  -.  ( A  =  C  \/  A  =  D )
) )
3 neanior 2427 . . . 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 451 . 2  |-  ( ( A  e.  B  /\  -.  A  e.  { C ,  D } )  <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D ) ) )
6 eldif 3130 . 2  |-  ( A  e.  ( B  \  { C ,  D }
)  <->  ( A  e.  B  /\  -.  A  e.  { C ,  D } ) )
7 3anass 977 . 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 703    /\ w3a 973    = wceq 1348    e. wcel 2141    =/= wne 2340    \ cdif 3118   {cpr 3584
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-un 3125  df-sn 3589  df-pr 3590
This theorem is referenced by:  rexdifpr  3611  rplogbval  13657
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