Theorem dfin4

Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of Mendelson p. 231. (Contributed by NM, 25-Nov-2003)

	Ref	Expression
Assertion	dfin4	$$⊢A\cap B=A\setminus \left(A\setminus B\right)$$

Proof

Step	Hyp	Ref	Expression
1		inss1	$$⊢A\cap B\subseteq A$$
2		dfss4	$$⊢A\cap B\subseteq A\leftrightarrow A\setminus \left(A\setminus \left(A\cap B\right)\right)=A\cap B$$
3	1 2	mpbi	$$⊢A\setminus \left(A\setminus \left(A\cap B\right)\right)=A\cap B$$
4		difin	$$⊢A\setminus \left(A\cap B\right)=A\setminus B$$
5	4	difeq2i	$$⊢A\setminus \left(A\setminus \left(A\cap B\right)\right)=A\setminus \left(A\setminus B\right)$$
6	3 5	eqtr3i	$$⊢A\cap B=A\setminus \left(A\setminus B\right)$$