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Axiom ax-un 4363
Description: Axiom of Union. An axiom of Intuitionistic 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 4365 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4366. A version using class notation is uniex 4367.

This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4057), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3745 should not be confused with the union of two classes df-un 3080. Their relationship is shown in unipr 3758. (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  setvar  z
2 vw . . . . . . 7  setvar  w
31, 2wel 1482 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  setvar  x
52, 4wel 1482 . . . . . 6  wff  w  e.  x
63, 5wa 103 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
76, 2wex 1469 . . . 4  wff  E. w
( z  e.  w  /\  w  e.  x
)
8 vy . . . . 5  setvar  y
91, 8wel 1482 . . . 4  wff  z  e.  y
107, 9wi 4 . . 3  wff  ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1110, 1wal 1330 . 2  wff  A. z
( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1211, 8wex 1469 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  4364  axun2  4365  bj-axun2  13284
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