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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 7479 | . . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) |
3 | elpwpw 5017 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) | |
4 | elpwb 4542 | . 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) | |
5 | 2, 3, 4 | 3bitr4i 305 | 1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 𝒫 cpw 4532 ∪ cuni 4831 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-pow 5259 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-pw 4534 df-uni 4832 |
This theorem is referenced by: elpwunicl 30304 |
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