Theorem eldifad 3208

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3206. (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 3206	. . 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 2200   \ cdif 3194

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199

This theorem is referenced by:  fimax2gtri  7072  finexdc  7073  elssdc  7075  unfidisj  7095  undifdc  7097  ssfirab  7109  fnfi  7114  iunfidisj  7124  dcfi  7159  hashunlem  11038  zfz1isolemiso  11074  fsumrelem  11998  fprodcl2lem  12132  fprodap0  12148  fprodrec  12156  fprodap0f  12163  fprodle  12167  iuncld  14805  fsumcncntop  15257  gausslemma2dlem0i  15752  gausslemma2dlem4  15759  gausslemma2dlem5a  15760  gausslemma2dlem7  15763  lgseisenlem1  15765  lgseisenlem2  15766  lgseisenlem3  15767  lgseisenlem4  15768  lgseisen  15769  lgsquadlem1  15772  lgsquadlem2  15773  lgsquadlem3  15774  bj-charfun  16253