Axiom ax-un 4526

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 4528 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4529. A version using class notation is uniex 4530.

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 4206), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 266).

The union of a class df-uni 3890 should not be confused with the union of two classes df-un 3202. Their relationship is shown in unipr 3903. (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

Step	Hyp	Ref	Expression
1		vz	. . . . . . 7 setvar z
2		vw	. . . . . . 7 setvar w
3	1, 2	wel 2201	. . . . . 6 wff z e. w
4		vx	. . . . . . 7 setvar x
5	2, 4	wel 2201	. . . . . 6 wff w e. x
6	3, 5	wa 104	. . . . 5 wff ( z e. w /\ w e. x )
7	6, 2	wex 1538	. . . 4 wff E. w ( z e. w /\ w e. x )
8		vy	. . . . 5 setvar y
9	1, 8	wel 2201	. . . 4 wff z e. y
10	7, 9	wi 4	. . 3 wff ( E. w ( z e. w /\ w e. x ) -> z e. y )
11	10, 1	wal 1393	. 2 wff A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
12	11, 8	wex 1538	1 wff E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )

This axiom is referenced by:  zfun  4527  axun2  4528  bj-axun2  16420