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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 4121. (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 2149 | . . . . 5 wff 𝑥 ∈ 𝑏 |
4 | va | . . . . . . 7 setvar 𝑎 | |
5 | 1, 4 | wel 2149 | . . . . . 6 wff 𝑥 ∈ 𝑎 |
6 | wph | . . . . . 6 wff 𝜑 | |
7 | 5, 6 | wa 104 | . . . . 5 wff (𝑥 ∈ 𝑎 ∧ 𝜑) |
8 | 3, 7 | wb 105 | . . . 4 wff (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
9 | 8, 1 | wal 1351 | . . 3 wff ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
10 | 9, 2 | wex 1492 | . 2 wff ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
11 | 10, 4 | wal 1351 | 1 wff ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
This axiom is referenced by: bdsep1 14607 |
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