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Mirrors > Home > ILE Home > Th. List > Mathboxes > ax-bdsep | GIF version |
Description: Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 4054. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
ax-bdsep.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
ax-bdsep | ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . . . 6 setvar 𝑥 | |
2 | vb | . . . . . 6 setvar 𝑏 | |
3 | 1, 2 | wel 1482 | . . . . 5 wff 𝑥 ∈ 𝑏 |
4 | va | . . . . . . 7 setvar 𝑎 | |
5 | 1, 4 | wel 1482 | . . . . . 6 wff 𝑥 ∈ 𝑎 |
6 | wph | . . . . . 6 wff 𝜑 | |
7 | 5, 6 | wa 103 | . . . . 5 wff (𝑥 ∈ 𝑎 ∧ 𝜑) |
8 | 3, 7 | wb 104 | . . . 4 wff (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
9 | 8, 1 | wal 1330 | . . 3 wff ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
10 | 9, 2 | wex 1469 | . 2 wff ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
11 | 10, 4 | wal 1330 | 1 wff ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
This axiom is referenced by: bdsep1 13254 |
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