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Mirrors > Home > MPE Home > Th. List > elpwi2 | Structured version Visualization version GIF version |
Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
elpwi2.1 | ⊢ 𝐵 ∈ 𝑉 |
elpwi2.2 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
elpwi2 | ⊢ 𝐴 ∈ 𝒫 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi2.2 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | elpwi2.1 | . . 3 ⊢ 𝐵 ∈ 𝑉 | |
3 | elpw2g 5238 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
5 | 1, 4 | mpbir 232 | 1 ⊢ 𝐴 ∈ 𝒫 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 |
This theorem is referenced by: canth 7100 aceq3lem 9534 axdc3lem4 9863 uzf 12234 ixxf 12736 fzf 12884 bitsf 15764 prdsval 16716 prdsds 16725 wunnat 17214 ocvfval 20738 leordtval2 21748 cnpfval 21770 iscnp2 21775 islly2 22020 xkotf 22121 alexsubALTlem4 22586 sszcld 23352 bndth 23489 ishtpy 23503 fpwrelmap 30395 ballotlem2 31645 satfrnmapom 32514 cover2 34870 clsk1indlem1 40273 sprsymrelfolem1 43531 |
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