Theorem elunnel2

Description: A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	elunnel2	$$⊢\left(A\in \left(B\cup C\right)\wedge \neg A\in C\right)\to A\in B$$

Proof

Step	Hyp	Ref	Expression
1		elun	$$⊢A\in \left(B\cup C\right)\leftrightarrow \left(A\in B\vee A\in C\right)$$
2	1	biimpi	$$⊢A\in \left(B\cup C\right)\to \left(A\in B\vee A\in C\right)$$
3	2	orcomd	$$⊢A\in \left(B\cup C\right)\to \left(A\in C\vee A\in B\right)$$
4	3	orcanai	$$⊢\left(A\in \left(B\cup C\right)\wedge \neg A\in C\right)\to A\in B$$