Theorem eldifeldifsn

Description: An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022)

	Ref	Expression
Assertion	eldifeldifsn	$$⊢\left(X\in A\wedge Y\in \left(B\setminus A\right)\right)\to Y\in \left(B\setminus \left\{X\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		snssi	$$⊢X\in A\to \left\{X\right\}\subseteq A$$
2	1	sscond	$$⊢X\in A\to B\setminus A\subseteq B\setminus \left\{X\right\}$$
3	2	sselda	$$⊢\left(X\in A\wedge Y\in \left(B\setminus A\right)\right)\to Y\in \left(B\setminus \left\{X\right\}\right)$$