Description: There is a nonempty class in an indexed collection B ( x ) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | iunn0 | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexcom4 | ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝐵 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) | |
| 2 | eliun | ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) | |
| 3 | 2 | exbii | ⊢ ( ∃ 𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
| 4 | 1 3 | bitr4i | ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝐵 ↔ ∃ 𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
| 5 | n0 | ⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝐵 ) | |
| 6 | 5 | rexbii | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝐵 ) |
| 7 | n0 | ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) | |
| 8 | 4 6 7 | 3bitr4i | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |