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

Axiom ax-un 4120
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 4122 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4123. A version using class notation is uniex 4124.

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 3852), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3555 should not be confused with the union of two classes df-un 2899. Their relationship is shown in unipr 3568. (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 1375 . . . . . 6 wff z w
4 vx . . . . . . 7 setvar x
52, 4wel 1375 . . . . . 6 wff w x
63, 5wa 97 . . . . 5 wff (z w w x)
76, 2wex 1362 . . . 4 wff w(z w w x)
8 vy . . . . 5 setvar y
91, 8wel 1375 . . . 4 wff z y
107, 9wi 4 . . 3 wff (w(z w w x) → z y)
1110, 1wal 1226 . 2 wff z(w(z w w x) → z y)
1211, 8wex 1362 1 wff yz(w(z w w x) → z y)
Colors of variables: wff set class
This axiom is referenced by:  zfun  4121  axun2  4122  bj-axun2  7138
  Copyright terms: Public domain W3C validator