Theorem axun2 4172

Description: A variant of the Axiom of Union ax-un 4170. 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 4170	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
2	1	bm1.3ii 3878	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-sep 3875  ax-un 4170

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)