Description: Obsolete version of pwunss as of 30-Dec-2023. (Contributed by NM, 23-Nov-2003) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | pwunssOLD | ⊢ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) ⊆ 𝒫 ( 𝐴 ∪ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssun | ⊢ ( ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) → 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) ) | |
| 2 | elun | ⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) ↔ ( 𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵 ) ) | |
| 3 | velpw | ⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) | |
| 4 | velpw | ⊢ ( 𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵 ) | |
| 5 | 3 4 | orbi12i | ⊢ ( ( 𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) ) |
| 6 | 2 5 | bitri | ⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) ) |
| 7 | velpw | ⊢ ( 𝑥 ∈ 𝒫 ( 𝐴 ∪ 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) ) | |
| 8 | 1 6 7 | 3imtr4i | ⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) → 𝑥 ∈ 𝒫 ( 𝐴 ∪ 𝐵 ) ) |
| 9 | 8 | ssriv | ⊢ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) ⊆ 𝒫 ( 𝐴 ∪ 𝐵 ) |