Theorem axnul 3903

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 3896. 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 3902).

This theorem should not be referenced by any proof. Instead, use ax-nul 3904 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 3896	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ ( y e. y /\ -. y e. y ) ) )
2		pm3.24 659	. . . . . 6 |- -. ( y e. y /\ -. y e. y )
3	2	intnan 871	. . . . 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 632	. . . 4 |- ( ( y e. x <-> ( y e. z /\ ( y e. y /\ -. y e. y ) ) ) -> -. y e. x )
6	5	alimi 1384	. . 3 |- ( A. y ( y e. x <-> ( y e. z /\ ( y e. y /\ -. y e. y ) ) ) -> A. y -. y e. x )
7	6	eximi 1531	. 2 |- ( E. x A. y ( y e. x <-> ( y e. z /\ ( y e. y /\ -. y e. y ) ) ) -> E. x A. y -. y e. x )
8	1, 7	ax-mp 7	1 |- E. x A. y -. y e. x

Syntax hints:   -. wn 3   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-sep 3896

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)