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 4550 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶) | |
3 | ssinss1 4213 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | inex1g 5215 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V) | |
6 | elpwg 4544 | . . 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 259 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 |
This theorem is referenced by: difelcarsg 31563 inelcarsg 31564 carsgclctunlem1 31570 carsgclctunlem2 31572 carsgclctunlem3 31573 carsgclctun 31574 |
Copyright terms: Public domain | W3C validator |