Theorem dfdif2 3161

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 3155	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
2		df-rab 2481	. 2 |- { x e. A | -. x e. B } = { x | ( x e. A /\ -. x e. B ) }
3	1, 2	eqtr4i 2217	1 |- ( A \ B ) = { x e. A | -. x e. B }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   \ cdif 3150

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-rab 2481  df-dif 3155

This theorem is referenced by:  dfdif3  3269  difeq1  3270  difeq2  3271  nfdif  3280  difidALT  3516