Theorem eldifad 3141

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3139. (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 3139	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 122	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simpld 112	1 |- ( ph -> A e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2148   \ cdif 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132

This theorem is referenced by:  fimax2gtri  6901  finexdc  6902  unfidisj  6921  undifdc  6923  ssfirab  6933  fnfi  6936  iunfidisj  6945  dcfi  6980  hashunlem  10784  zfz1isolemiso  10819  fsumrelem  11479  fprodcl2lem  11613  fprodap0  11629  fprodrec  11637  fprodap0f  11644  fprodle  11648  iuncld  13618  fsumcncntop  14059  lgseisenlem1  14453  lgseisenlem2  14454  bj-charfun  14562