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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 4046. (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 1481 | . . . . 5 wff 𝑥 ∈ 𝑏 |
4 | va | . . . . . . 7 setvar 𝑎 | |
5 | 1, 4 | wel 1481 | . . . . . 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 1329 | . . 3 wff ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
10 | 9, 2 | wex 1468 | . 2 wff ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
11 | 10, 4 | wal 1329 | 1 wff ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
This axiom is referenced by: bdsep1 13083 |
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