Theorem eldifd 3139

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3138. (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 3138	. 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 3126

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 2739  df-dif 3131

This theorem is referenced by:  exmidundif  4204  exmidundifim  4205  frirrg  4348  dcdifsnid  6500  phpelm  6861  findcard2d  6886  findcard2sd  6887  diffifi  6889  unsnfidcex  6914  unsnfidcel  6915  undifdcss  6917  difinfsnlem  7093  difinfsn  7094  hashunlem  10775  seq3coll  10813  fsum3cvg  11377  isumss  11390  fisumss  11391  fproddccvg  11571  fprodssdc  11589  sqrt2irr0  12154  nnoddn2prmb  12252  logbgcd1irr  14167