Metamath Proof Explorer


Theorem n0f

Description: A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. This version of n0 requires only that x not be free in, rather than not occur in, A . (Contributed by NM, 17-Oct-2003)

Ref Expression
Hypothesis eq0f.1 _xA
Assertion n0f AxxA

Proof

Step Hyp Ref Expression
1 eq0f.1 _xA
2 df-ne A¬A=
3 1 neq0f ¬A=xxA
4 2 3 bitri AxxA