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Theorem axun2 4235
Description: A variant of the Axiom of Union ax-un 4233. 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 4233 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
21bm1.3ii 3934 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 102    <-> wb 103   A.wal 1285   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-sep 3931  ax-un 4233
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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