![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 3875. (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 1394 | . . . . 5 wff 𝑥 ∈ 𝑏 |
4 | va | . . . . . . 7 setvar 𝑎 | |
5 | 1, 4 | wel 1394 | . . . . . 6 wff 𝑥 ∈ 𝑎 |
6 | wph | . . . . . 6 wff 𝜑 | |
7 | 5, 6 | wa 97 | . . . . 5 wff (𝑥 ∈ 𝑎 ∧ 𝜑) |
8 | 3, 7 | wb 98 | . . . 4 wff (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
9 | 8, 1 | wal 1241 | . . 3 wff ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
10 | 9, 2 | wex 1381 | . 2 wff ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
11 | 10, 4 | wal 1241 | 1 wff ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
This axiom is referenced by: bdsep1 10005 |
Copyright terms: Public domain | W3C validator |