Theorem axnul 2709

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 2703. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tells us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 2707).

This proof, suggested by Jeff Hoffman (3-Feb-2008), does not require the set existence axiom ax-9 965, unlike some empty set existence proofs (such as the remark in [Kunen] p. 11). However, it uses ax-4 973, which also presupposes the existence of at least one set, i.e it does not hold in a "free logic" valid in empty domains. Theorem ax4 972, which shows the redundancy of ax-4 973, depends on ax-9 965 for its proof. See axnul2 2708 for a similar proof directly from ax-rep 2693.

This theorem should not be referenced by any proof. Instead, use ax-nul 2710 below so that the uses of the Null Set Axiom can be more easily identified.

Assertion

Ref	Expression
axnul	|- E.xA.y -. y e. x

Distinct variable group:   x,y

Proof of Theorem axnul

Step	Hyp	Ref	Expression
1		ax-sep 2703	. 2 |- E.xA.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y)))
2		pm3.24 658	. . . . . 6 |- -. (y e. y /\ -. y e. y)
3	2	intnan 691	. . . . 5 |- -. (y e. z /\ (y e. y /\ -. y e. y))
4		id 59	. . . . 5 |- ((y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> (y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))))
5	3, 4	mtbiri 717	. . . 4 |- ((y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> -. y e. x)
6	5	19.20i 992	. . 3 |- (A.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> A.y -. y e. x)
7	6	19.22i 1040	. 2 |- (E.xA.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> E.xA.y -. y e. x)
8	1, 7	ax-mp 7	1 |- E.xA.y -. y e. x

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980

This theorem is referenced by:  snex 2750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981

Copyright terms: Public domain