Metamath Proof Explorer


Theorem eldifbd

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif . (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypothesis eldifbd.1
|- ( ph -> A e. ( B \ C ) )
Assertion eldifbd
|- ( ph -> -. A e. C )

Proof

Step Hyp Ref Expression
1 eldifbd.1
 |-  ( ph -> A e. ( B \ C ) )
2 eldif
 |-  ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3 1 2 sylib
 |-  ( ph -> ( A e. B /\ -. A e. C ) )
4 3 simprd
 |-  ( ph -> -. A e. C )