Theorem elsn

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. (Contributed by NM, 13-Sep-1995)

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

Proof

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