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Mathbox for Thierry Arnoux |
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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 4506 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶) | |
3 | ssinss1 4164 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | inex1g 5187 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V) | |
6 | elpwg 4500 | . . 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 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 |
This theorem is referenced by: difelcarsg 31678 inelcarsg 31679 carsgclctunlem1 31685 carsgclctunlem2 31687 carsgclctunlem3 31688 carsgclctun 31689 |
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