Theorem n0moeu

Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010)

	Ref	Expression
Assertion	n0moeu	$$⊢A\ne \varnothing \to \left({\exists }^{*}xx\in A\leftrightarrow \exists !xx\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
2	1	biimpi	$$⊢A\ne \varnothing \to \exists xx\in A$$
3	2	biantrurd	$$⊢A\ne \varnothing \to \left({\exists }^{*}xx\in A\leftrightarrow \left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)\right)$$
4		df-eu	$$⊢\exists !xx\in A\leftrightarrow \left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)$$
5	3 4	bitr4di	$$⊢A\ne \varnothing \to \left({\exists }^{*}xx\in A\leftrightarrow \exists !xx\in A\right)$$