Description: The Null Set Axiom of ZF set theory: there exists a set with no
elements. Axiom of Empty Set of Enderton p. 18. In some textbooks,
this is presented as a separate axiom; here we show it can be derived
from Separation ax-sep . This version of the Null Set Axiom tells us
that at least one empty set exists, but does not tell us that it is
unique - we need the Axiom of Extensionality to do that (see nulmo ).

This proof, suggested by Jeff Hoffman, uses only ax-4 and ax-gen from predicate calculus, which are valid in "free logic" i.e. logic
holding in an empty domain (see Axiom A5 and Rule R2 of LeBlanc
p. 277). Thus, our ax-sep implies the existence of at least one set.
Note that Kunen's version of ax-sep (Axiom 3 of Kunen p. 11) does
not imply the existence of a set because his is universally closed,
i.e., prefixed with universal quantifiers to eliminate all free
variables. His existence is provided by a separate axiom stating
E. x x = x (Axiom 0 of Kunen p. 10).

This theorem should not be referenced by any proof. Instead, use
ax-nul below so that the uses of the Null Set Axiom can be more easily
identified. (Contributed by Jeff Hoffman, 3-Feb-2008)(Revised by NM, 4-Feb-2008)(New usage is discouraged.)(Proof modification is discouraged.)