Theorem elunnel1

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

	Ref	Expression
Assertion	elunnel1	$$⊢\left(A\in \left(B\cup C\right)\wedge \neg A\in B\right)\to A\in C$$

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	orcanai	$$⊢\left(A\in \left(B\cup C\right)\wedge \neg A\in B\right)\to A\in C$$