Theorem eqsnd

Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023) (Proof shortened by SN, 3-Jul-2025)

	Ref	Expression
Hypotheses	eqsnd.1	$$⊢\left(\phi \wedge x\in A\right)\to x=B$$
	eqsnd.2	$$⊢\phi \to B\in A$$
Assertion	eqsnd	$$⊢\phi \to A=\left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		eqsnd.1	$$⊢\left(\phi \wedge x\in A\right)\to x=B$$
2		eqsnd.2	$$⊢\phi \to B\in A$$
3	1	ralrimiva	$$⊢\phi \to \forall x\in Ax=B$$
4	2	ne0d	$$⊢\phi \to A\ne \varnothing$$
5		eqsn	$$⊢A\ne \varnothing \to \left(A=\left\{B\right\}\leftrightarrow \forall x\in Ax=B\right)$$
6	4 5	syl	$$⊢\phi \to \left(A=\left\{B\right\}\leftrightarrow \forall x\in Ax=B\right)$$
7	3 6	mpbird	$$⊢\phi \to A=\left\{B\right\}$$