Theorem axnul 4788

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 4781. 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 4786).

This proof, suggested by Jeff Hoffman, uses only ax-4 1737 and ax-gen 1722 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 4781 implies the existence of at least one set. Note that Kunen's version of ax-sep 4781 (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).

See axnulALT 4787 for a proof directly from ax-rep 4771.

This theorem should not be referenced by any proof. Instead, use ax-nul 4789 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	|- E. x A. y -. y e. x

Distinct variable group:   x, y

Proof of Theorem axnul

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 4781	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ F. ) )
2		fal 1490	. . . . 5 |- -. F.
3	2	intnan 960	. . . 4 |- -. ( y e. z /\ F. )
4		id 22	. . . 4 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x <-> ( y e. z /\ F. ) ) )
5	3, 4	mtbiri 317	. . 3 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> -. y e. x )
6	5	alimi 1739	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y -. y e. x )
7	1, 6	eximii 1764	1 |- E. x A. y -. y e. x

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   A.wal 1481   F. wfal 1488   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705

This theorem is referenced by: (None)