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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 3438 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | rabexgfGS.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | dfssf 3967 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)) |
4 | rabidim1 3440 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mpgbir 1793 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
6 | elex 3480 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
7 | ssexg 5324 | . 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ V) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | |
8 | 5, 6, 7 | sylancr 585 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Ⅎwnfc 2875 {crab 3418 Vcvv 3461 ⊆ wss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-rab 3419 df-v 3463 df-in 3951 df-ss 3961 |
This theorem is referenced by: abrexexd 32382 |
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