Theorem eusnsn

Description: There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022)

	Ref	Expression
Assertion	eusnsn	$$⊢\exists !x\left\{x\right\}=\left\{y\right\}$$

Proof

Step	Hyp	Ref	Expression
1		equequ2	$$⊢z=y\to \left(x=z\leftrightarrow x=y\right)$$
2	1	bibi2d	$$⊢z=y\to \left(\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=z\right)\leftrightarrow \left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=y\right)\right)$$
3	2	albidv	$$⊢z=y\to \left(\forall x\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=z\right)\leftrightarrow \forall x\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=y\right)\right)$$
4		sneqbg	$$⊢x\in \mathrm{V}\to \left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=y\right)$$
5	4	elv	$$⊢\left\{x\right\}=\left\{y\right\}\leftrightarrow x=y$$
6	5	ax-gen	$$⊢\forall x\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=y\right)$$
7	3 6	speivw	$$⊢\exists z\forall x\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=z\right)$$
8		eu6	$$⊢\exists !x\left\{x\right\}=\left\{y\right\}\leftrightarrow \exists z\forall x\left(\left\{x\right\}=\left\{y\right\}\leftrightarrow x=z\right)$$
9	7 8	mpbir	$$⊢\exists !x\left\{x\right\}=\left\{y\right\}$$