Theorem brdif 4147

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 3210	. 2 |- ( <. A , B >. e. ( R \ S ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
2		df-br 4094	. 2 |- ( A ( R \ S ) B <-> <. A , B >. e. ( R \ S ) )
3		df-br 4094	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4094	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	4	notbii 674	. . 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 2202   \ cdif 3198   <.cop 3676   class class class wbr 4093

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-br 4094

This theorem is referenced by:  fundif  5381  fndmdif  5761  brdifun  6772