Theorem axun2 6951

Description: A variant of the Axiom of Union ax-un 6949. 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 6949	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
2	1	bm1.3ii 4784	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-11 2034  ax-12 2047  ax-13 2246  ax-sep 4781  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)