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Theorem axun2 4397
Description: A variant of the Axiom of Union ax-un 4395. 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 4395 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
21bm1.3ii 4087 1  |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1333   E.wex 1472
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-14 2131  ax-sep 4084  ax-un 4395
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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