Theorem brdif 4097

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 3175	. 2 |- ( <. A , B >. e. ( R \ S ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
2		df-br 4045	. 2 |- ( A ( R \ S ) B <-> <. A , B >. e. ( R \ S ) )
3		df-br 4045	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4045	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	4	notbii 670	. . 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 2176   \ cdif 3163   <.cop 3636   class class class wbr 4044

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-br 4045

This theorem is referenced by:  fundif  5318  fndmdif  5685  brdifun  6647