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 7489 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) | |
3 | 1, 2 | sylib 220 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 𝒫 cpw 4539 ∪ cuni 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-pow 5266 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4839 |
This theorem is referenced by: ldgenpisyslem1 31422 |
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