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Metamath Proof Explorer

Theorem n0rex

Description: There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020)

		Ref	Expression
	Assertion	n0rex	$⊢ A \neq \emptyset \to \exists x \in A x \in A$

Step	Hyp	Ref	Expression
1		id	$⊢ x \in A \to x \in A$
2	1	ancli	$⊢ x \in A \to (x \in A \land x \in A)$
3	2	eximi	$⊢ \exists x x \in A \to \exists x (x \in A \land x \in A)$
4		n0	$⊢ A \neq \emptyset \leftrightarrow \exists x x \in A$
5		df-rex	$⊢ \exists x \in A x \in A \leftrightarrow \exists x (x \in A \land x \in A)$
6	3 4 5	3imtr4i	$⊢ A \neq \emptyset \to \exists x \in A x \in A$