Theorem eldifad 3077

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3075. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifad.1	|- ( ph -> A e. ( B \ C ) )

Assertion

Ref	Expression
eldifad	|- ( ph -> A e. B )

Proof of Theorem eldifad

Step	Hyp	Ref	Expression
1		eldifad.1	. . 3 |- ( ph -> A e. ( B \ C ) )
2		eldif 3075	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 121	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simpld 111	1 |- ( ph -> A e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 1480   \ cdif 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068

This theorem is referenced by:  fimax2gtri  6788  finexdc  6789  unfidisj  6803  undifdc  6805  ssfirab  6815  fnfi  6818  iunfidisj  6827  hashunlem  10543  zfz1isolemiso  10575  fsumrelem  11233  iuncld  12273  fsumcncntop  12714