Theorem eldifad 3168

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3166. (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 3166	. . 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 2167   \ cdif 3154

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159

This theorem is referenced by:  fimax2gtri  6963  finexdc  6964  unfidisj  6984  undifdc  6986  ssfirab  6998  fnfi  7003  iunfidisj  7013  dcfi  7048  hashunlem  10898  zfz1isolemiso  10933  fsumrelem  11638  fprodcl2lem  11772  fprodap0  11788  fprodrec  11796  fprodap0f  11803  fprodle  11807  iuncld  14361  fsumcncntop  14813  gausslemma2dlem0i  15308  gausslemma2dlem4  15315  gausslemma2dlem5a  15316  gausslemma2dlem7  15319  lgseisenlem1  15321  lgseisenlem2  15322  lgseisenlem3  15323  lgseisenlem4  15324  lgseisen  15325  lgsquadlem1  15328  lgsquadlem2  15329  lgsquadlem3  15330  bj-charfun  15463