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

Axiom ax-un 4350
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 4352 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4353. A version using class notation is uniex 4354.

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 4044), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3732 should not be confused with the union of two classes df-un 3070. Their relationship is shown in unipr 3745. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-un 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
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 1481 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 1481 . . . . . 6 wff 𝑤𝑥
63, 5wa 103 . . . . 5 wff (𝑧𝑤𝑤𝑥)
76, 2wex 1468 . . . 4 wff 𝑤(𝑧𝑤𝑤𝑥)
8 vy . . . . 5 setvar 𝑦
91, 8wel 1481 . . . 4 wff 𝑧𝑦
107, 9wi 4 . . 3 wff (∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1110, 1wal 1329 . 2 wff 𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1211, 8wex 1468 1 wff 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff set class
This axiom is referenced by:  zfun  4351  axun2  4352  bj-axun2  13102
  Copyright terms: Public domain W3C validator