Theorem snex

Description: A singleton is a set. Theorem 7.12 of Quine p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT . (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 19-May-2013)

	Ref	Expression
Assertion	snex	⊢ { 𝐴 } ∈ V

Proof

Step	Hyp	Ref	Expression
1		snexg	⊢ ( 𝐴 ∈ V → { 𝐴 } ∈ V )
2		snprc	⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
3	2	biimpi	⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } = ∅ )
4		0ex	⊢ ∅ ∈ V
5	3 4	eqeltrdi	⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } ∈ V )
6	1 5	pm2.61i	⊢ { 𝐴 } ∈ V