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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwimpALT | Structured version Visualization version GIF version |
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpALT 43774 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 43774. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 43418 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 9 used elpwgded 43413). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 5 used elpwi 4609). (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspwimpALT | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3478 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝑥 ∈ V) |
3 | id 22 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴) | |
4 | elpwi 4609 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) |
6 | id 22 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | sylan9ssr 3996 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵) |
8 | 2, 7 | elpwgded 43413 | . . . . . 6 ⊢ ((⊤ ∧ (𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵) |
9 | 8 | uunT1 43629 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵) |
10 | 9 | ex 413 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) |
11 | 10 | alrimiv 1930 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) |
12 | dfss2 3968 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) | |
13 | 12 | biimpri 227 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
14 | 11, 13 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
15 | 14 | idiALT 43326 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 ⊤wtru 1542 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 𝒫 cpw 4602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-pw 4604 |
This theorem is referenced by: (None) |
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