Theorem eqsnd

Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023)

	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		simpr	$$⊢\left(\phi \wedge x=B\right)\to x=B$$
4	2	adantr	$$⊢\left(\phi \wedge x=B\right)\to B\in A$$
5	3 4	eqeltrd	$$⊢\left(\phi \wedge x=B\right)\to x\in A$$
6	1 5	impbida	$$⊢\phi \to \left(x\in A\leftrightarrow x=B\right)$$
7		velsn	$$⊢x\in \left\{B\right\}\leftrightarrow x=B$$
8	6 7	syl6bbr	$$⊢\phi \to \left(x\in A\leftrightarrow x\in \left\{B\right\}\right)$$
9	8	eqrdv	$$⊢\phi \to A=\left\{B\right\}$$