Theorem indif

Description: Intersection with class difference. Theorem 34 of Suppes p. 29. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	indif	$$⊢A\cap \left(A\setminus B\right)=A\setminus B$$

Proof

Step	Hyp	Ref	Expression
1		dfin4	$$⊢A\cap \left(A\setminus B\right)=A\setminus \left(A\setminus \left(A\setminus B\right)\right)$$
2		dfin4	$$⊢A\cap B=A\setminus \left(A\setminus B\right)$$
3	2	difeq2i	$$⊢A\setminus \left(A\cap B\right)=A\setminus \left(A\setminus \left(A\setminus B\right)\right)$$
4		difin	$$⊢A\setminus \left(A\cap B\right)=A\setminus B$$
5	1 3 4	3eqtr2i	$$⊢A\cap \left(A\setminus B\right)=A\setminus B$$