Description: Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rabexgfGS.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| Assertion | rabexgfGS | ⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabexgfGS.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| 2 | nfrab1 | ⊢ Ⅎ 𝑥 { 𝑥 ∈ 𝐴 ∣ 𝜑 } | |
| 3 | 2 1 | dfss2f | ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → 𝑥 ∈ 𝐴 ) ) |
| 4 | rabidim1 | ⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → 𝑥 ∈ 𝐴 ) | |
| 5 | 3 4 | mpgbir | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴 |
| 6 | elex | ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) | |
| 7 | ssexg | ⊢ ( ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴 ∧ 𝐴 ∈ V ) → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V ) | |
| 8 | 5 6 7 | sylancr | ⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V ) |