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Theorem eldifbd 3114
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3111. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
eldifbd.1  |-  ( ph  ->  A  e.  ( B 
\  C ) )
Assertion
Ref Expression
eldifbd  |-  ( ph  ->  -.  A  e.  C
)

Proof of Theorem eldifbd
StepHypRef Expression
1 eldifbd.1 . . 3  |-  ( ph  ->  A  e.  ( B 
\  C ) )
2 eldif 3111 . . 3  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
31, 2sylib 121 . 2  |-  ( ph  ->  ( A  e.  B  /\  -.  A  e.  C
) )
43simprd 113 1  |-  ( ph  ->  -.  A  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 2128    \ cdif 3099
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-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104
This theorem is referenced by:  fidifsnen  6815  fiunsnnn  6826  fimax2gtri  6846  unfidisj  6866  ssfirab  6878  fnfi  6881  iunfidisj  6890  hashunlem  10678  hashxp  10700  zfz1isolemiso  10710  fsumconst  11351  fsumrelem  11368  fprodcl2lem  11502  fprodconst  11517  fprodap0  11518  fprodrec  11526  fprodap0f  11533  fprodle  11537  fprodmodd  11538  fsumcncntop  12956  bj-charfun  13382  bj-charfundc  13383
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