Theorem eldifd 3141

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3140. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
eldifd.1	|- ( ph -> A e. B )
eldifd.2	|- ( ph -> -. A e. C )

Assertion

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

Proof of Theorem eldifd

Step	Hyp	Ref	Expression
1		eldifd.1	. 2 |- ( ph -> A e. B )
2		eldifd.2	. 2 |- ( ph -> -. A e. C )
3		eldif 3140	. 2 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
4	1, 2, 3	sylanbrc 417	1 |- ( ph -> A e. ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2148   \ cdif 3128

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133

This theorem is referenced by:  exmidundif  4208  exmidundifim  4209  frirrg  4352  dcdifsnid  6507  phpelm  6868  findcard2d  6893  findcard2sd  6894  diffifi  6896  unsnfidcex  6921  unsnfidcel  6922  undifdcss  6924  difinfsnlem  7100  difinfsn  7101  hashunlem  10786  seq3coll  10824  fsum3cvg  11388  isumss  11401  fisumss  11402  fproddccvg  11582  fprodssdc  11600  sqrt2irr0  12166  nnoddn2prmb  12264  logbgcd1irr  14424