Metamath Proof Explorer


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