| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > elpwpwel | Structured version Visualization version GIF version | ||
| Description: A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.) |
| Ref | Expression |
|---|---|
| elpwpwel | ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniexb 7784 | . . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
| 2 | 1 | anbi1i 624 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) |
| 3 | elpwpw 5102 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) | |
| 4 | elpwb 4608 | . 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) | |
| 5 | 2, 3, 4 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 |
| 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 ax-pow 5365 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-uni 4908 |
| This theorem is referenced by: elpwunicl 32567 |
| Copyright terms: Public domain | W3C validator |