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Metamath Proof Explorer

Theorem neldif

Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994)

		Ref	Expression
	Assertion	neldif	$⊢ (A \in B \land \neg A \in (B ∖ C)) \to A \in C$

Step	Hyp	Ref	Expression
1		eldif	$⊢ A \in (B ∖ C) \leftrightarrow (A \in B \land \neg A \in C)$
2	1	simplbi2	$⊢ A \in B \to (\neg A \in C \to A \in (B ∖ C))$
3	2	con1d	$⊢ A \in B \to (\neg A \in (B ∖ C) \to A \in C)$
4	3	imp	$⊢ (A \in B \land \neg A \in (B ∖ C)) \to A \in C$