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Mirrors > Home > MPE Home > Th. List > abssexg | Structured version Visualization version 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 5384 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | df-pw 4607 | . . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
3 | 2 | eleq1i 2830 | . . 3 ⊢ (𝒫 𝐴 ∈ V ↔ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) |
4 | simpl 482 | . . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝑥 ⊆ 𝐴) | |
5 | 4 | ss2abi 4077 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} |
6 | ssexg 5329 | . . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∧ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V) | |
7 | 5, 6 | mpan 690 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V) |
8 | 3, 7 | sylbi 217 | . 2 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V) |
9 | 1, 8 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 |
This theorem is referenced by: pmex 8870 tgval 22978 |
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