| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rabexgfGS | Structured version Visualization version GIF version | ||
| 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 |
|---|---|
| rabexgfGS.1 | ⊢ Ⅎ𝑥𝐴 |
| Ref | Expression |
|---|---|
| rabexgfGS | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfrab1 3456 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
| 2 | rabexgfGS.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 1, 2 | dfssf 3973 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)) |
| 4 | rabidim1 3458 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
| 5 | 3, 4 | mpgbir 1799 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
| 6 | elex 3500 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 7 | ssexg 5321 | . 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ V) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | |
| 8 | 5, 6, 7 | sylancr 587 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Ⅎwnfc 2889 {crab 3435 Vcvv 3479 ⊆ wss 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 |
| This theorem is referenced by: abrexexd 32517 |
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