Theorem axnul 3444

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 3438. 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 3442).

This proof, suggested by Jeff Hoffman (3-Feb-2008), does not require the set existence axiom ax-9 1307, unlike some empty set existence proofs (such as the remark in [Kunen] p. 11). However, it uses ax-4 1319, 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 1318, which shows the redundancy of ax-4 1319, depends on ax-9 1307 for its proof. See axnulALT 3443 for a similar proof directly from ax-rep 3428.

This theorem should not be referenced by any proof (except snex 3492 below). Instead, use ax-nul 3445 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 3438	. 2 |- E.xA.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y)))
2		pm3.24 720	. . . . . 6 |- -. (y e. y /\ -. y e. y)
3	2	intnan 755	. . . . 5 |- -. (y e. z /\ (y e. y /\ -. y e. y))
4		id 73	. . . . 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 785	. . . 4 |- ((y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> -. y e. x)
6	5	alimi 1338	. . 3 |- (A.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> A.y -. y e. x)
7	6	eximi 1387	. 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 163   /\ wa 240  A.wal 1296   e. wcel 1300  E.wex 1326

This theorem is referenced by:  snexOLD 3493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327

Copyright terms: Public domain