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Mirrors > Home > ILE Home > Th. List > ax-un | GIF version |
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 4466 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 4467. A version using class
notation is uniex 4468.
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 4150), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 266). The union of a class df-uni 3836 should not be confused with the union of two classes df-un 3157. Their relationship is shown in unipr 3849. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
ax-un | ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vz | . . . . . . 7 setvar 𝑧 | |
2 | vw | . . . . . . 7 setvar 𝑤 | |
3 | 1, 2 | wel 2165 | . . . . . 6 wff 𝑧 ∈ 𝑤 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 2, 4 | wel 2165 | . . . . . 6 wff 𝑤 ∈ 𝑥 |
6 | 3, 5 | wa 104 | . . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
7 | 6, 2 | wex 1503 | . . . 4 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
8 | vy | . . . . 5 setvar 𝑦 | |
9 | 1, 8 | wel 2165 | . . . 4 wff 𝑧 ∈ 𝑦 |
10 | 7, 9 | wi 4 | . . 3 wff (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
11 | 10, 1 | wal 1362 | . 2 wff ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
12 | 11, 8 | wex 1503 | 1 wff ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
Colors of variables: wff set class |
This axiom is referenced by: zfun 4465 axun2 4466 bj-axun2 15407 |
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