Theorem eqeuel

Description: A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022)

	Ref	Expression
Assertion	eqeuel	$$⊢\left(A\ne \varnothing \wedge \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\right)\to \exists !xx\in A$$

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	anim1i	$$⊢\left(A\ne \varnothing \wedge \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\right)\to \left(\exists xx\in A\wedge \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\right)$$
4		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
5	4	eu4	$$⊢\exists !xx\in A\leftrightarrow \left(\exists xx\in A\wedge \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\right)$$
6	3 5	sylibr	$$⊢\left(A\ne \varnothing \wedge \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\right)\to \exists !xx\in A$$