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Metamath Proof Explorer

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

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	orcomd	$⊢ A \in (B \cup C) \to (A \in C \lor A \in B)$
4	3	orcanai	$⊢ (A \in (B \cup C) \land \neg A \in C) \to A \in B$