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Mirrors > Home > ILE Home > Th. List > abssexg | GIF version |
Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
abssexg | ⊢ (A ∈ 𝑉 → {x ∣ (x ⊆ A ∧ φ)} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 3924 | . 2 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V) | |
2 | df-pw 3353 | . . . 4 ⊢ 𝒫 A = {x ∣ x ⊆ A} | |
3 | 2 | eleq1i 2100 | . . 3 ⊢ (𝒫 A ∈ V ↔ {x ∣ x ⊆ A} ∈ V) |
4 | simpl 102 | . . . . 5 ⊢ ((x ⊆ A ∧ φ) → x ⊆ A) | |
5 | 4 | ss2abi 3006 | . . . 4 ⊢ {x ∣ (x ⊆ A ∧ φ)} ⊆ {x ∣ x ⊆ A} |
6 | ssexg 3887 | . . . 4 ⊢ (({x ∣ (x ⊆ A ∧ φ)} ⊆ {x ∣ x ⊆ A} ∧ {x ∣ x ⊆ A} ∈ V) → {x ∣ (x ⊆ A ∧ φ)} ∈ V) | |
7 | 5, 6 | mpan 400 | . . 3 ⊢ ({x ∣ x ⊆ A} ∈ V → {x ∣ (x ⊆ A ∧ φ)} ∈ V) |
8 | 3, 7 | sylbi 114 | . 2 ⊢ (𝒫 A ∈ V → {x ∣ (x ⊆ A ∧ φ)} ∈ V) |
9 | 1, 8 | syl 14 | 1 ⊢ (A ∈ 𝑉 → {x ∣ (x ⊆ A ∧ φ)} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 {cab 2023 Vcvv 2551 ⊆ wss 2911 𝒫 cpw 3351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 |
This theorem is referenced by: (None) |
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