Theorem eldifad 3177

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3175. (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 3175	. . 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 2176   \ cdif 3163

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168

This theorem is referenced by:  fimax2gtri  7000  finexdc  7001  unfidisj  7021  undifdc  7023  ssfirab  7035  fnfi  7040  iunfidisj  7050  dcfi  7085  hashunlem  10951  zfz1isolemiso  10986  fsumrelem  11815  fprodcl2lem  11949  fprodap0  11965  fprodrec  11973  fprodap0f  11980  fprodle  11984  iuncld  14620  fsumcncntop  15072  gausslemma2dlem0i  15567  gausslemma2dlem4  15574  gausslemma2dlem5a  15575  gausslemma2dlem7  15578  lgseisenlem1  15580  lgseisenlem2  15581  lgseisenlem3  15582  lgseisenlem4  15583  lgseisen  15584  lgsquadlem1  15587  lgsquadlem2  15588  lgsquadlem3  15589  bj-charfun  15780