Theorem inindif

Description: The intersection and class difference of a class with another class are disjoint. With inundif , this shows that such intersection and class difference partition the class A . (Contributed by Thierry Arnoux, 13-Sep-2017)

	Ref	Expression
Assertion	inindif	$$⊢\left(A\cap C\right)\cap \left(A\setminus C\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		inss2	$$⊢A\cap C\subseteq C$$
2		ssinss1	$$⊢A\cap C\subseteq C\to \left(A\cap C\right)\cap A\subseteq C$$
3	1 2	ax-mp	$$⊢\left(A\cap C\right)\cap A\subseteq C$$
4		inssdif0	$$⊢\left(A\cap C\right)\cap A\subseteq C\leftrightarrow \left(A\cap C\right)\cap \left(A\setminus C\right)=\varnothing$$
5	3 4	mpbi	$$⊢\left(A\cap C\right)\cap \left(A\setminus C\right)=\varnothing$$