Theorem axun2 4413

Description: A variant of the Axiom of Union ax-un 4411. 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. (Contributed by NM, 4-Jun-2006.)

Assertion

Ref	Expression
axun2	|- E. y A. 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 4411	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
2	1	bm1.3ii 4103	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-14 2139  ax-sep 4100  ax-un 4411

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)