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Axiom ax-un 4197
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 4199 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4200. A version using class notation is uniex 4201.

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 3905), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 257).

The union of a class df-uni 3608 should not be confused with the union of two classes df-un 2949. Their relationship is shown in unipr 3621. (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 1410 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  setvar  x
52, 4wel 1410 . . . . . 6  wff  w  e.  x
63, 5wa 101 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
76, 2wex 1397 . . . 4  wff  E. w
( z  e.  w  /\  w  e.  x
)
8 vy . . . . 5  setvar  y
91, 8wel 1410 . . . 4  wff  z  e.  y
107, 9wi 4 . . 3  wff  ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1110, 1wal 1257 . 2  wff  A. z
( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1211, 8wex 1397 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  4198  axun2  4199  bj-axun2  10401
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