Theorem zfnul 1746

Description: Another version of the Null Set axiom, expressed in terms of logical primitives and proved directly from Aussonderung. Axiom of Empty Set of [Enderton] p. 18.

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem zfnul

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 |- z e. V
2	1	zfaus 1480	. 2 |- E.xA.y(y e. x <-> (y e. z /\ -. y e. z))
3		pm3.24 496	. . . . 5 |- -. (y e. z /\ -. y e. z)
4		id 9	. . . . 5 |- ((y e. x <-> (y e. z /\ -. y e. z)) -> (y e. x <-> (y e. z /\ -. y e. z)))
5	3, 4	mtbiri 539	. . . 4 |- ((y e. x <-> (y e. z /\ -. y e. z)) -> -. y e. x)
6	5	19.20i 691	. . 3 |- (A.y(y e. x <-> (y e. z /\ -. y e. z)) -> A.y -. y e. x)
7	6	19.22i 723	. 2 |- (E.xA.y(y e. x <-> (y e. z /\ -. y e. z)) -> E.xA.y -. y e. x)
8	2, 7	ax-mp 6	1 |- E.xA.y -. y e. x

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wel 803

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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