Theorem elpsseu

Description: A class has a unique element just in case it has a unique proper subset. (Contributed by Adrian Ducourtial, 26-Oct-2026)

	Ref	Expression
Assertion	elpsseu	$$⊢\exists !xx\in A\leftrightarrow \exists !yy\subset A$$

Proof

Step	Hyp	Ref	Expression
1		elpssex	$$⊢\exists xx\in A\leftrightarrow \exists yy\subset A$$
2		elpssmo	$$⊢{\exists }^{*}xx\in A\leftrightarrow {\exists }^{*}yy\subset A$$
3	1 2	anbi12i	$$⊢\left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)\leftrightarrow \left(\exists yy\subset A\wedge {\exists }^{*}yy\subset A\right)$$
4		df-eu	$$⊢\exists !xx\in A\leftrightarrow \left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)$$
5		df-eu	$$⊢\exists !yy\subset A\leftrightarrow \left(\exists yy\subset A\wedge {\exists }^{*}yy\subset A\right)$$
6	3 4 5	3bitr4i	$$⊢\exists !xx\in A\leftrightarrow \exists !yy\subset A$$