Theorem snprc

Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of Quine p. 48. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	snprc	$$⊢\neg A\in \mathrm{V}\leftrightarrow \left\{A\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		velsn	$$⊢x\in \left\{A\right\}\leftrightarrow x=A$$
2	1	exbii	$$⊢\exists xx\in \left\{A\right\}\leftrightarrow \exists xx=A$$
3		neq0	$$⊢\neg \left\{A\right\}=\varnothing \leftrightarrow \exists xx\in \left\{A\right\}$$
4		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
5	2 3 4	3bitr4i	$$⊢\neg \left\{A\right\}=\varnothing \leftrightarrow A\in \mathrm{V}$$
6	5	con1bii	$$⊢\neg A\in \mathrm{V}\leftrightarrow \left\{A\right\}=\varnothing$$