Description: A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | fzon0 | ⊢ ( ( 𝑀 ..^ 𝑁 ) ≠ ∅ ↔ 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 | ⊢ ( ( 𝑀 ..^ 𝑁 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝑀 ..^ 𝑁 ) ) | |
2 | elfzolt3b | ⊢ ( 𝑥 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ) | |
3 | 2 | exlimiv | ⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ) |
4 | 1 3 | sylbi | ⊢ ( ( 𝑀 ..^ 𝑁 ) ≠ ∅ → 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ) |
5 | ne0i | ⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀 ..^ 𝑁 ) ≠ ∅ ) | |
6 | 4 5 | impbii | ⊢ ( ( 𝑀 ..^ 𝑁 ) ≠ ∅ ↔ 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ) |