Theorem sneqr

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of TakeutiZaring p. 15. (Contributed by NM, 27-Aug-1993)

	Ref	Expression
Hypothesis	sneqr.1	$$⊢A\in \mathrm{V}$$
Assertion	sneqr	$$⊢\left\{A\right\}=\left\{B\right\}\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		sneqr.1	$$⊢A\in \mathrm{V}$$
2		sneqrg	$$⊢A\in \mathrm{V}\to \left(\left\{A\right\}=\left\{B\right\}\to A=B\right)$$
3	1 2	ax-mp	$$⊢\left\{A\right\}=\left\{B\right\}\to A=B$$