Theorem eldifbd 2930

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2927. (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 2927	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 127	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simprd 107	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   e. wcel 1393   \ cdif 2914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920

This theorem is referenced by:  fidifsnen  6331  fiunsnnn  6338