Metamath Proof Explorer

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	$⊢ (A \in (B \cup C) \land \neg A \in B) \to A \in C$

Proof

Step	Hyp	Ref	Expression
1		elun	$⊢ A \in (B \cup C) \leftrightarrow (A \in B \lor A \in C)$
2	1	biimpi	$⊢ A \in (B \cup C) \to (A \in B \lor A \in C)$
3	2	orcanai	$⊢ (A \in (B \cup C) \land \neg A \in B) \to A \in C$