Theorem axnul 4580

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 4573. 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 zfnuleu 4578).

This proof, suggested by Jeff Hoffman, uses only ax-4 1631 and ax-gen 1618 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 4573 implies the existence of at least one set. Note that Kunen's version of ax-sep 4573 (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 (Axiom 0 of [Kunen] p. 10).

See axnulALT 4579 for a proof directly from ax-rep 4563.

This theorem should not be referenced by any proof. Instead, use ax-nul 4581 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.)

Assertion

Ref	Expression
axnul

Distinct variable group:   ,

Proof of Theorem axnul

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 4573	. 2
2		pm3.24 882	. . . . . 6
3	2	intnan 914	. . . . 5
4		id 22	. . . . 5
5	3, 4	mtbiri 303	. . . 4
6	5	alimi 1633	. . 3
7	6	eximi 1656	. 2
8	1, 7	ax-mp 5	1

Syntax hints:  -.wn 3  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613