Theorem eldifbd 3178

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3175. (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 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	simprd 114	1 |- ( ph -> -. A e. C )

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:  fidifsnen  6969  fiunsnnn  6980  fimax2gtri  7000  unfidisj  7021  ssfirab  7035  fnfi  7040  iunfidisj  7050  hashunlem  10951  hashxp  10973  zfz1isolemiso  10986  fsumconst  11798  fsumrelem  11815  fprodcl2lem  11949  fprodconst  11964  fprodap0  11965  fprodrec  11973  fprodap0f  11980  fprodle  11984  fprodmodd  11985  fsumcncntop  15072  bj-charfun  15780  bj-charfundc  15781