Theorem eldifbd 2985

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2982. (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 2982	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 120	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simprd 112	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   e. wcel 1433   \ cdif 2970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975

This theorem is referenced by:  fidifsnen  6355  fiunsnnn  6365  fnfi  6388