Theorem eldifbd 3212

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3209. (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 3209	. . 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 2202   \ cdif 3197

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202

This theorem is referenced by:  fidifsnen  7057  fiunsnnn  7070  fimax2gtri  7091  unfidisj  7114  ssfirab  7129  fnfi  7135  iunfidisj  7145  hashunlem  11067  hashxp  11090  zfz1isolemiso  11103  fsumconst  12016  fsumrelem  12033  fprodcl2lem  12167  fprodconst  12182  fprodap0  12183  fprodrec  12191  fprodap0f  12198  fprodle  12202  fprodmodd  12203  fsumcncntop  15293  1loopgrvd0fi  16159  bj-charfun  16405  bj-charfundc  16406  gfsumcl  16690