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Axiom ax-un 4170
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 4172 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4173. A version using class notation is uniex 4174.

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 3878), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3581 should not be confused with the union of two classes df-un 2922. Their relationship is shown in unipr 3594. (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 1394 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  setvar  x
52, 4wel 1394 . . . . . 6  wff  w  e.  x
63, 5wa 97 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
76, 2wex 1381 . . . 4  wff  E. w
( z  e.  w  /\  w  e.  x
)
8 vy . . . . 5  setvar  y
91, 8wel 1394 . . . 4  wff  z  e.  y
107, 9wi 4 . . 3  wff  ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1110, 1wal 1241 . 2  wff  A. z
( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
1211, 8wex 1381 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  4171  axun2  4172  bj-axun2  10035
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