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Mirrors > Home > MPE Home > Th. List > rabelpw | Structured version Visualization version GIF version |
Description: A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.) |
Ref | Expression |
---|---|
rabelpw | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3993 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | elpw2g 5237 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 261 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {crab 3065 ⊆ wss 3866 𝒫 cpw 4513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-in 3873 df-ss 3883 df-pw 4515 |
This theorem is referenced by: satfvsuclem2 33035 |
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