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	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\in A\forall y\in Ax=y$$

Proof

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