Theorem eldifbd 3141

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3138. (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 3138	. . 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 2148   \ cdif 3126

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131

This theorem is referenced by:  fidifsnen  6869  fiunsnnn  6880  fimax2gtri  6900  unfidisj  6920  ssfirab  6932  fnfi  6935  iunfidisj  6944  hashunlem  10779  hashxp  10801  zfz1isolemiso  10814  fsumconst  11457  fsumrelem  11474  fprodcl2lem  11608  fprodconst  11623  fprodap0  11624  fprodrec  11632  fprodap0f  11639  fprodle  11643  fprodmodd  11644  fsumcncntop  13949  bj-charfun  14441  bj-charfundc  14442