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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rabexf | Structured version Visualization version GIF version | ||
| Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| rabexf.1 | ⊢ Ⅎ𝑥𝐴 | 
| rabexf.2 | ⊢ 𝐴 ∈ 𝑉 | 
| Ref | Expression | 
|---|---|
| rabexf | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabexf.2 | . 2 ⊢ 𝐴 ∈ 𝑉 | |
| 2 | rabexf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | rabexgf 45029 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | 
| 4 | 1, 3 | ax-mp 5 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 Vcvv 3480 | 
| 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 2708 ax-sep 5296 | 
| 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 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 | 
| This theorem is referenced by: limsupequzmpt2 45733 liminfequzmpt2 45806 fsupdm 46857 finfdm 46861 | 
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