Theorem eldifbd 3169

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3166. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifbd.1	|- ( ph -> A e. ( B \ C ) )

Assertion

Ref	Expression
eldifbd	|- ( ph -> -. A e. C )

Proof of Theorem eldifbd

Step	Hyp	Ref	Expression
1		eldifbd.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	simprd 114	1 |- ( ph -> -. A e. C )

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:  fidifsnen  6940  fiunsnnn  6951  fimax2gtri  6971  unfidisj  6992  ssfirab  7006  fnfi  7011  iunfidisj  7021  hashunlem  10915  hashxp  10937  zfz1isolemiso  10950  fsumconst  11638  fsumrelem  11655  fprodcl2lem  11789  fprodconst  11804  fprodap0  11805  fprodrec  11813  fprodap0f  11820  fprodle  11824  fprodmodd  11825  fsumcncntop  14911  bj-charfun  15561  bj-charfundc  15562