Theorem brdif 4695

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

Assertion

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

Proof of Theorem brdif

Step	Hyp	Ref	Expression
1		eldif 3222	. 2 |- (<.A, B>. e. (R \ S) <-> (<.A, B>. e. R /\ -. <.A, B>. e. S))
2		df-br 4641	. 2 |- (A(R \ S)B <-> <.A, B>. e. (R \ S))
3		df-br 4641	. . 3 |- (ARB <-> <.A, B>. e. R)
4		df-br 4641	. . . 4 |- (ASB <-> <.A, B>. e. S)
5	4	notbii 287	. . 3 |- (-. ASB <-> -. <.A, B>. e. S)
6	3, 5	anbi12i 678	. 2 |- ((ARB /\ -. ASB) <-> (<.A, B>. e. R /\ -. <.A, B>. e. S))
7	1, 2, 6	3bitr4i 268	1 |- (A(R \ S)B <-> (ARB /\ -. ASB))

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358   e. wcel 1710   \ cdif 3207  <.cop 4562   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-br 4641

This theorem is referenced by:  fnfullfunlem1  5857  brltc  6115