Theorem eldifad 3113

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3111. (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 3111	. . 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 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:  fimax2gtri  6847  finexdc  6848  unfidisj  6867  undifdc  6869  ssfirab  6879  fnfi  6882  iunfidisj  6891  dcfi  6926  hashunlem  10682  zfz1isolemiso  10714  fsumrelem  11372  fprodcl2lem  11506  fprodap0  11522  fprodrec  11530  fprodap0f  11537  fprodle  11541  iuncld  12557  fsumcncntop  12998  bj-charfun  13424