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Theorem eldifbd 3083
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3080. (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 3080 . . 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 1480    \ cdif 3068
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073
This theorem is referenced by:  fidifsnen  6764  fiunsnnn  6775  fimax2gtri  6795  unfidisj  6810  ssfirab  6822  fnfi  6825  iunfidisj  6834  hashunlem  10550  hashxp  10572  zfz1isolemiso  10582  fsumconst  11223  fsumrelem  11240  fsumcncntop  12725
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