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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 4607 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶) | |
| 3 | ssinss1 4246 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| 5 | inex1g 5319 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V) | |
| 6 | elpwg 4603 | . . 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 257 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: difelcarsg 34312 inelcarsg 34313 carsgclctunlem1 34319 carsgclctunlem2 34321 carsgclctunlem3 34322 carsgclctun 34323 |
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