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Mirrors > Home > ILE Home > Th. List > ax-coll | GIF version |
Description: Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3916 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.) |
Ref | Expression |
---|---|
ax-coll.1 | ⊢ Ⅎ𝑏φ |
Ref | Expression |
---|---|
ax-coll | ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . . 4 wff φ | |
2 | vy | . . . 4 setvar y | |
3 | 1, 2 | wex 1378 | . . 3 wff ∃yφ |
4 | vx | . . 3 setvar x | |
5 | va | . . . 4 setvar 𝑎 | |
6 | 5 | cv 1241 | . . 3 class 𝑎 |
7 | 3, 4, 6 | wral 2300 | . 2 wff ∀x ∈ 𝑎 ∃yφ |
8 | vb | . . . . . 6 setvar 𝑏 | |
9 | 8 | cv 1241 | . . . . 5 class 𝑏 |
10 | 1, 2, 9 | wrex 2301 | . . . 4 wff ∃y ∈ 𝑏 φ |
11 | 10, 4, 6 | wral 2300 | . . 3 wff ∀x ∈ 𝑎 ∃y ∈ 𝑏 φ |
12 | 11, 8 | wex 1378 | . 2 wff ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ |
13 | 7, 12 | wi 4 | 1 wff (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) |
Colors of variables: wff set class |
This axiom is referenced by: repizf 3864 bnd 3916 |
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