Theorem elsn2

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. This variation requires only that B , rather than A , be a set. (Contributed by NM, 12-Jun-1994)

	Ref	Expression
Hypothesis	elsn2.1	$$⊢B\in \mathrm{V}$$
Assertion	elsn2	$$⊢A\in \left\{B\right\}\leftrightarrow A=B$$

Proof

Step	Hyp	Ref	Expression
1		elsn2.1	$$⊢B\in \mathrm{V}$$
2		elsn2g	$$⊢B\in \mathrm{V}\to \left(A\in \left\{B\right\}\leftrightarrow A=B\right)$$
3	1 2	ax-mp	$$⊢A\in \left\{B\right\}\leftrightarrow A=B$$