Theorem eldifbd 3166

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3163. (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 3163	. . 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 2164   \ cdif 3151

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156

This theorem is referenced by:  fidifsnen  6928  fiunsnnn  6939  fimax2gtri  6959  unfidisj  6980  ssfirab  6992  fnfi  6997  iunfidisj  7007  hashunlem  10878  hashxp  10900  zfz1isolemiso  10913  fsumconst  11600  fsumrelem  11617  fprodcl2lem  11751  fprodconst  11766  fprodap0  11767  fprodrec  11775  fprodap0f  11782  fprodle  11786  fprodmodd  11787  fsumcncntop  14746  bj-charfun  15369  bj-charfundc  15370