Theorem dfdif2 3583

Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
dfdif2	|- ( A \ B ) = { x e. A | -. x e. B }

Distinct variable groups:   x, A   x, B

Proof of Theorem dfdif2

Step	Hyp	Ref	Expression
1		df-dif 3577	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
2		df-rab 2921	. 2 |- { x e. A | -. x e. B } = { x | ( x e. A /\ -. x e. B ) }
3	1, 2	eqtr4i 2647	1 |- ( A \ B ) = { x e. A | -. x e. B }

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-rab 2921  df-dif 3577

This theorem is referenced by:  difeq1  3721  difeq2  3722  nfdif  3731  difidALT  3949  ordintdif  5774  kmlem3  8974  incexc2  14570  cnambfre  33458