Theorem axnul 3873

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 3866. 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 3872).

This theorem should not be referenced by any proof. Instead, use ax-nul 3874 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.xA.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 3866	. 2 |- E.xA.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y)))
2		pm3.24 626	. . . . . 6 |- -. (y e. y /\ -. y e. y)
3	2	intnan 837	. . . . 5 |- -. (y e. z /\ (y e. y /\ -. y e. y))
4		id 19	. . . . 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 599	. . . 4 |- ((y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> -. y e. x)
6	5	alimi 1341	. . 3 |- (A.y(y e. x <-> (y e. z /\ (y e. y /\ -. y e. y))) -> A.y -. y e. x)
7	6	eximi 1488	. 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 3   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378   e. wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-sep 3866

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)