Axiom ax-un 4405

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 4407 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4408. A version using class notation is uniex 4409.

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 4097), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3784 should not be confused with the union of two classes df-un 3115. Their relationship is shown in unipr 3797. (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 2136	. . . . . 6 wff z e. w
4		vx	. . . . . . 7 setvar x
5	2, 4	wel 2136	. . . . . 6 wff w e. x
6	3, 5	wa 103	. . . . 5 wff ( z e. w /\ w e. x )
7	6, 2	wex 1479	. . . 4 wff E. w ( z e. w /\ w e. x )
8		vy	. . . . 5 setvar y
9	1, 8	wel 2136	. . . 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 1340	. 2 wff A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
12	11, 8	wex 1479	1 wff E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )

This axiom is referenced by:  zfun  4406  axun2  4407  bj-axun2  13632