Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 4542 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶) | |
3 | ssinss1 4171 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | inex1g 5243 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V) | |
6 | elpwg 4536 | . . 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 256 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-in 3894 df-ss 3904 df-pw 4535 |
This theorem is referenced by: difelcarsg 32277 inelcarsg 32278 carsgclctunlem1 32284 carsgclctunlem2 32286 carsgclctunlem3 32287 carsgclctun 32288 |
Copyright terms: Public domain | W3C validator |