Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | 1open.1 | ⊢ 𝑋 = ∪ 𝐽 | |
| Assertion | topopn | ⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1open.1 | ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | ssid | ⊢ 𝐽 ⊆ 𝐽 | |
| 3 | uniopn | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽 ) → ∪ 𝐽 ∈ 𝐽 ) | |
| 4 | 2 3 | mpan2 | ⊢ ( 𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽 ) |
| 5 | 1 4 | eqeltrid | ⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 ) |