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		19.21v	$$⊢\forall y\left(x\in A\to \left(y\in A\to x=y\right)\right)\leftrightarrow \left(x\in A\to \forall y\left(y\in A\to x=y\right)\right)$$
2		impexp	$$⊢\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \left(x\in A\to \left(y\in A\to x=y\right)\right)$$
3	2	albii	$$⊢\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \forall y\left(x\in A\to \left(y\in A\to x=y\right)\right)$$
4		df-ral	$$⊢\forall y\in Ax=y\leftrightarrow \forall y\left(y\in A\to x=y\right)$$
5	4	imbi2i	$$⊢\left(x\in A\to \forall y\in Ax=y\right)\leftrightarrow \left(x\in A\to \forall y\left(y\in A\to x=y\right)\right)$$
6	1 3 5	3bitr4i	$$⊢\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \left(x\in A\to \forall y\in Ax=y\right)$$
7	6	albii	$$⊢\forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \forall x\left(x\in A\to \forall y\in Ax=y\right)$$
8		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
9	8	mo4	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)$$
10		df-ral	$$⊢\forall x\in A\forall y\in Ax=y\leftrightarrow \forall x\left(x\in A\to \forall y\in Ax=y\right)$$
11	7 9 10	3bitr4i	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\in A\forall y\in Ax=y$$