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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwincl1 | Structured version Visualization version GIF version |
Description: Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
Ref | Expression |
---|---|
elpwincl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) |
Ref | Expression |
---|---|
elpwincl1 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwincl.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) | |
2 | elpwi 4498 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶) | |
3 | ssinss1 4129 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | inex1g 5188 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V) | |
6 | elpwg 4492 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶)) | |
7 | 1, 5, 6 | 3syl 18 | . 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶)) |
8 | 4, 7 | mpbird 260 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2113 Vcvv 3398 ∩ cin 3843 ⊆ wss 3844 𝒫 cpw 4489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-sep 5168 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-in 3851 df-ss 3861 df-pw 4491 |
This theorem is referenced by: difelcarsg 31847 inelcarsg 31848 carsgclctunlem1 31854 carsgclctunlem2 31856 carsgclctunlem3 31857 carsgclctun 31858 |
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