Theorem dfdif2 3207

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 3201	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
2		df-rab 2518	. 2 |- { x e. A | -. x e. B } = { x | ( x e. A /\ -. x e. B ) }
3	1, 2	eqtr4i 2254	1 |- ( A \ B ) = { x e. A | -. x e. B }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1397   e. wcel 2201   {cab 2216   {crab 2513   \ cdif 3196

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-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-cleq 2223  df-rab 2518  df-dif 3201

This theorem is referenced by:  dfdif3  3316  difeq1  3317  difeq2  3318  nfdif  3327  difidALT  3563