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.) |
Ref | Expression |
---|---|
elpwunicl.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
elpwunicl.2 | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵) |
Ref | Expression |
---|---|
elpwunicl | ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwunicl.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵) | |
2 | elpwg 4541 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵)) |
4 | 1, 3 | mpbid 233 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝐵) |
5 | pwuniss 30233 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝐵) |
7 | uniexg 7456 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → ∪ 𝐴 ∈ V) | |
8 | elpwg 4541 | . . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)) | |
9 | 1, 7, 8 | 3syl 18 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)) |
10 | 6, 9 | mpbird 258 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-sn 4558 df-pr 4560 df-uni 4831 |
This theorem is referenced by: ldgenpisyslem1 31321 |
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