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Mirrors > Home > MPE Home > Th. List > elpwi2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of elpwi2 5209 as of 26-May-2024. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpwi2.1 | ⊢ 𝐵 ∈ 𝑉 |
elpwi2.2 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
elpwi2OLD | ⊢ 𝐴 ∈ 𝒫 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi2.2 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | elpwi2.1 | . . 3 ⊢ 𝐵 ∈ 𝑉 | |
3 | elpw2g 5207 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
5 | 1, 4 | mpbir 234 | 1 ⊢ 𝐴 ∈ 𝒫 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2112 ⊆ wss 3854 𝒫 cpw 4487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 ax-sep 5162 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-in 3861 df-ss 3871 df-pw 4489 |
This theorem is referenced by: (None) |
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