Theorem eldifbd 3209

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3206. (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 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	simprd 114	1 |- ( ph -> -. A e. C )

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:  fidifsnen  7045  fiunsnnn  7056  fimax2gtri  7077  unfidisj  7100  ssfirab  7114  fnfi  7119  iunfidisj  7129  hashunlem  11043  hashxp  11066  zfz1isolemiso  11079  fsumconst  11986  fsumrelem  12003  fprodcl2lem  12137  fprodconst  12152  fprodap0  12153  fprodrec  12161  fprodap0f  12168  fprodle  12172  fprodmodd  12173  fsumcncntop  15262  bj-charfun  16279  bj-charfundc  16280