Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of Enderton p. 33. (Contributed by NM, 25-Nov-2003)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | iunpwss | ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssiun | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥 ) | |
| 2 | eliun | ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ) | |
| 3 | velpw | ⊢ ( 𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥 ) | |
| 4 | 3 | rexbii | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 ) |
| 5 | 2 4 | bitri | ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 ) |
| 6 | velpw | ⊢ ( 𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴 ) | |
| 7 | uniiun | ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 8 | 7 | sseq2i | ⊢ ( 𝑦 ⊆ ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥 ) |
| 9 | 6 8 | bitri | ⊢ ( 𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥 ) |
| 10 | 1 5 9 | 3imtr4i | ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴 ) |
| 11 | 10 | ssriv | ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 |