ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-un Structured version   GIF version

Axiom ax-un 4093
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 4095 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4096. A version using class notation is uniex 4097.

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 3830), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3533 should not be confused with the union of two classes df-un 2900. Their relationship is shown in unipr 3546. (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 1376 . . . . . 6 wff z w
4 vx . . . . . . 7 setvar x
52, 4wel 1376 . . . . . 6 wff w x
63, 5wa 97 . . . . 5 wff (z w w x)
76, 2wex 1361 . . . 4 wff w(z w w x)
8 vy . . . . 5 setvar y
91, 8wel 1376 . . . 4 wff z y
107, 9wi 4 . . 3 wff (w(z w w x) → z y)
1110, 1wal 1314 . 2 wff z(w(z w w x) → z y)
1211, 8wex 1361 1 wff yz(w(z w w x) → z y)
Colors of variables: wff set class
This axiom is referenced by:  zfun  4094  axun2  4095
  Copyright terms: Public domain W3C validator