Theorem brdif 4058

Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Assertion

Ref	Expression
brdif	|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )

Proof of Theorem brdif

Step	Hyp	Ref	Expression
1		eldif 3140	. 2 |- ( <. A , B >. e. ( R \ S ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
2		df-br 4006	. 2 |- ( A ( R \ S ) B <-> <. A , B >. e. ( R \ S ) )
3		df-br 4006	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4006	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	4	notbii 668	. . 3 |- ( -. A S B <-> -. <. A , B >. e. S )
6	3, 5	anbi12i 460	. 2 |- ( ( A R B /\ -. A S B ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
7	1, 2, 6	3bitr4i 212	1 |- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   e. wcel 2148   \ cdif 3128   <.cop 3597   class class class wbr 4005

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  df-br 4006

This theorem is referenced by:  fndmdif  5623  brdifun  6564