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Axiom ax-un 4109
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 4111 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4112. A version using class notation is uniex 4113.

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 3842), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3545 should not be confused with the union of two classes df-un 2891. Their relationship is shown in unipr 3558. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-un yz(w(z w w x) → z 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 1368 . . . . . 6 wff z w
4 vx . . . . . . 7 setvar x
52, 4wel 1368 . . . . . 6 wff w x
63, 5wa 97 . . . . 5 wff (z w w x)
76, 2wex 1355 . . . 4 wff w(z w w x)
8 vy . . . . 5 setvar y
91, 8wel 1368 . . . 4 wff z y
107, 9wi 4 . . 3 wff (w(z w w x) → z y)
1110, 1wal 1222 . 2 wff z(w(z w w x) → z y)
1211, 8wex 1355 1 wff yz(w(z w w x) → z y)
Colors of variables: wff set class
This axiom is referenced by:  zfun  4110  axun2  4111  bj-axun2  8368
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