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 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html . It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 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 ). 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 ). (Contributed by Alan Sare, 3-Dec-2015) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sspwimpALT | ⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex | ⊢ 𝑥 ∈ V | |
| 2 | 1 | a1i | ⊢ ( ⊤ → 𝑥 ∈ V ) |
| 3 | id | ⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴 ) | |
| 4 | elpwi | ⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴 ) | |
| 5 | 3 4 | syl | ⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴 ) |
| 6 | id | ⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵 ) | |
| 7 | 5 6 | sylan9ssr | ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) → 𝑥 ⊆ 𝐵 ) |
| 8 | 2 7 | elpwgded | ⊢ ( ( ⊤ ∧ ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) ) → 𝑥 ∈ 𝒫 𝐵 ) |
| 9 | 8 | uunT1 | ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) → 𝑥 ∈ 𝒫 𝐵 ) |
| 10 | 9 | ex | ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) ) |
| 11 | 10 | alrimiv | ⊢ ( 𝐴 ⊆ 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) ) |
| 12 | dfss2 | ⊢ ( 𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) ) | |
| 13 | 12 | biimpri | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) → 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |
| 14 | 11 13 | syl | ⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |
| 15 | 14 | idiALT | ⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |