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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sselpwuni | Structured version Visualization version GIF version | ||
| Description: Quantitative version of ssexg 5323: 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 5323 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
| 2 | ssuni 4932 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ ∪ 𝑉) | |
| 3 | 1, 2 | elpwd 4606 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-uni 4908 |
| This theorem is referenced by: bj-unirel 37052 |
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