Theorem mosneq

Description: There exists at most one set whose singleton is equal to a given class. See also moeq . (Contributed by BJ, 24-Sep-2022)

	Ref	Expression
Assertion	mosneq	$$⊢{\exists }^{*}x\left\{x\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		eqtr3	$$⊢\left(\left\{x\right\}=A\wedge \left\{y\right\}=A\right)\to \left\{x\right\}=\left\{y\right\}$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	sneqr	$$⊢\left\{x\right\}=\left\{y\right\}\to x=y$$
4	1 3	syl	$$⊢\left(\left\{x\right\}=A\wedge \left\{y\right\}=A\right)\to x=y$$
5	4	gen2	$$⊢\forall x\forall y\left(\left(\left\{x\right\}=A\wedge \left\{y\right\}=A\right)\to x=y\right)$$
6		sneq	$$⊢x=y\to \left\{x\right\}=\left\{y\right\}$$
7	6	eqeq1d	$$⊢x=y\to \left(\left\{x\right\}=A\leftrightarrow \left\{y\right\}=A\right)$$
8	7	mo4	$$⊢{\exists }^{*}x\left\{x\right\}=A\leftrightarrow \forall x\forall y\left(\left(\left\{x\right\}=A\wedge \left\{y\right\}=A\right)\to x=y\right)$$
9	5 8	mpbir	$$⊢{\exists }^{*}x\left\{x\right\}=A$$