Theorem dfdif2 3129

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 3123	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
2		df-rab 2457	. 2 |- { x e. A | -. x e. B } = { x | ( x e. A /\ -. x e. B ) }
3	1, 2	eqtr4i 2194	1 |- ( A \ B ) = { x e. A | -. x e. B }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   \ cdif 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-rab 2457  df-dif 3123

This theorem is referenced by:  dfdif3  3237  difeq1  3238  difeq2  3239  nfdif  3248  difidALT  3484