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Metamath Proof Explorer

Theorem moel

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018) Avoid ax-11 . (Revised by Wolf Lammen, 23-Nov-2024)

		Ref	Expression
	Assertion	moel	$⊢ \exists^{*} x x \in A \leftrightarrow \forall x \in A \forall y \in A x = y$

Step	Hyp	Ref	Expression
1		eleq1w	$⊢ x = y \to (x \in A \leftrightarrow y \in A)$
2	1	mo4	$⊢ \exists^{*} x x \in A \leftrightarrow \forall x \forall y ((x \in A \land y \in A) \to x = y)$
3		r2al	$⊢ \forall x \in A \forall y \in A x = y \leftrightarrow \forall x \forall y ((x \in A \land y \in A) \to x = y)$
4	2 3	bitr4i	$⊢ \exists^{*} x x \in A \leftrightarrow \forall x \in A \forall y \in A x = y$