Theorem axun2 2924

Description: A variant of the Axiom of Union ax-un 2922. For any set x, there exists a set y whose members are exactly the members of the members of x i.e. the union of x. Axiom Union of [BellMachover] p. 466.

Assertion

Ref	Expression
axun2	|- E.yA.z(z e. y <-> E.w(z e. w /\ w e. x))

Distinct variable group:   x,w,y,z

Proof of Theorem axun2

Step	Hyp	Ref	Expression
1		ax-un 2922	. 2 |- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
2	1	bm1.3ii 2761	1 |- E.yA.z(z e. y <-> E.w(z e. w /\ w e. x))

Syntax hints:   <-> wb 153   /\ wa 230  A.wal 995   e. wcel 999  E.wex 1021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-sep 2758  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022

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