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Axiom ax-un 4512
Description: Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set  y exists that includes the union of a given set  x i.e. the collection of all members of the members of  x. The variant axun2 4514 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4515. A version using class notation is uniex 4516.

The union of a class df-uni 3828 should not be confused with the union of two classes df-un 3157. Their relationship is shown in unipr 3841. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-un  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
Distinct variable group:    x, w, y, z

Detailed syntax breakdown of Axiom ax-un
StepHypRef Expression
1 vz . . . . . . 7  set  z
2 vw . . . . . . 7  set  w
31, 2wel 1685 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1685 . . . . . 6  wff  w  e.  x
63, 5wa 358 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
76, 2wex 1528 . . . 4  wff  E. w
( z  e.  w  /\  w  e.  x
)
8 vy . . . . 5  set  y
91, 8wel 1685 . . . 4  wff  z  e.  y
107, 9wi 4 . . 3  wff  ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1110, 1wal 1527 . 2  wff  A. z
( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1211, 8wex 1528 1  wff  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
Colors of variables: wff set class
This axiom is referenced by:  zfun  4513  axun2  4514
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