Axiom ax-un 4236

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 4238 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4239. A version using class notation is uniex 4240.

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 3937), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262).

The union of a class df-uni 3639 should not be confused with the union of two classes df-un 2992. Their relationship is shown in unipr 3652. (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 1437	. . . . . 6 wff z e. w
4		vx	. . . . . . 7 setvar x
5	2, 4	wel 1437	. . . . . 6 wff w e. x
6	3, 5	wa 102	. . . . 5 wff ( z e. w /\ w e. x )
7	6, 2	wex 1424	. . . 4 wff E. w ( z e. w /\ w e. x )
8		vy	. . . . 5 setvar y
9	1, 8	wel 1437	. . . 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 1285	. 2 wff A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
12	11, 8	wex 1424	1 wff E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )

This axiom is referenced by:  zfun  4237  axun2  4238  bj-axun2  11275