Theorem eldifad 3087

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifad.1	|- ( ph -> A e. ( B \ C ) )

Assertion

Ref	Expression
eldifad	|- ( ph -> A e. B )

Proof of Theorem eldifad

Step	Hyp	Ref	Expression
1		eldifad.1	. . 3 |- ( ph -> A e. ( B \ C ) )
2		eldif 3085	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 121	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simpld 111	1 |- ( ph -> A e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 1481   \ cdif 3073

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078

This theorem is referenced by:  fimax2gtri  6803  finexdc  6804  unfidisj  6818  undifdc  6820  ssfirab  6830  fnfi  6833  iunfidisj  6842  hashunlem  10582  zfz1isolemiso  10614  fsumrelem  11272  iuncld  12323  fsumcncntop  12764