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Metamath Proof Explorer

Theorem untint

Description: If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011)

		Ref	Expression
	Assertion	untint	$⊢ \exists x \in A \forall y \in x \neg y \in y \to \forall y \in ⋂ A \neg y \in y$

Proof

Step	Hyp	Ref	Expression
1		intss1	$⊢ x \in A \to ⋂ A \subseteq x$
2		ssralv	$⊢ ⋂ A \subseteq x \to (\forall y \in x \neg y \in y \to \forall y \in ⋂ A \neg y \in y)$
3	1 2	syl	$⊢ x \in A \to (\forall y \in x \neg y \in y \to \forall y \in ⋂ A \neg y \in y)$
4	3	rexlimiv	$⊢ \exists x \in A \forall y \in x \neg y \in y \to \forall y \in ⋂ A \neg y \in y$