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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwunicl | Structured version Visualization version GIF version |
Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.) |
Ref | Expression |
---|---|
elpwunicl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵) |
Ref | Expression |
---|---|
elpwunicl | ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwunicl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵) | |
2 | elpwpwel 7767 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) | |
3 | 1, 2 | sylib 217 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 𝒫 cpw 4598 ∪ cuni 4903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-pow 5359 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rab 3420 df-v 3465 df-in 3946 df-ss 3956 df-pw 4600 df-uni 4904 |
This theorem is referenced by: ldgenpisyslem1 33839 |
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