Theorem eldifbd 3186

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3183. (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 3183	. . 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 2178   \ cdif 3171

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176

This theorem is referenced by:  fidifsnen  6993  fiunsnnn  7004  fimax2gtri  7024  unfidisj  7045  ssfirab  7059  fnfi  7064  iunfidisj  7074  hashunlem  10986  hashxp  11008  zfz1isolemiso  11021  fsumconst  11880  fsumrelem  11897  fprodcl2lem  12031  fprodconst  12046  fprodap0  12047  fprodrec  12055  fprodap0f  12062  fprodle  12066  fprodmodd  12067  fsumcncntop  15154  bj-charfun  15942  bj-charfundc  15943