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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sselpwuni | Structured version Visualization version GIF version |
Description: Quantitative version of ssexg 5318: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.) |
Ref | Expression |
---|---|
bj-sselpwuni | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5318 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
2 | ssuni 4930 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ ∪ 𝑉) | |
3 | 1, 2 | elpwd 4604 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3940 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 |
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-rab 3420 df-v 3465 df-in 3947 df-ss 3957 df-pw 4600 df-uni 4904 |
This theorem is referenced by: bj-unirel 36586 |
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