Theorem eldifbd 3143

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3140. (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 3140	. . 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 3128

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 2741  df-dif 3133

This theorem is referenced by:  fidifsnen  6872  fiunsnnn  6883  fimax2gtri  6903  unfidisj  6923  ssfirab  6935  fnfi  6938  iunfidisj  6947  hashunlem  10786  hashxp  10808  zfz1isolemiso  10821  fsumconst  11464  fsumrelem  11481  fprodcl2lem  11615  fprodconst  11630  fprodap0  11631  fprodrec  11639  fprodap0f  11646  fprodle  11650  fprodmodd  11651  fsumcncntop  14095  bj-charfun  14598  bj-charfundc  14599