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Theorem axun2 4451
Description: A variant of the Axiom of Union ax-un 4449. 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
StepHypRef Expression
1 ax-un 4449 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
21bm1.3ii 4084 1  |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537    e. wcel 1621
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-14 1626  ax-17 1628  ax-9 1684  ax-4 1692  ax-sep 4081  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540
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