Description: A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | iocnct.a | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
iocnct.b | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | ||
iocnct.l | ⊢ ( 𝜑 → 𝐴 < 𝐵 ) | ||
iocnct.c | ⊢ 𝐶 = ( 𝐴 (,] 𝐵 ) | ||
Assertion | iocnct | ⊢ ( 𝜑 → ¬ 𝐶 ≼ ω ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocnct.a | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
2 | iocnct.b | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | |
3 | iocnct.l | ⊢ ( 𝜑 → 𝐴 < 𝐵 ) | |
4 | iocnct.c | ⊢ 𝐶 = ( 𝐴 (,] 𝐵 ) | |
5 | eqid | ⊢ ( 𝐴 (,) 𝐵 ) = ( 𝐴 (,) 𝐵 ) | |
6 | 1 2 3 5 | ioonct | ⊢ ( 𝜑 → ¬ ( 𝐴 (,) 𝐵 ) ≼ ω ) |
7 | ioossioc | ⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,] 𝐵 ) | |
8 | 7 4 | sseqtrri | ⊢ ( 𝐴 (,) 𝐵 ) ⊆ 𝐶 |
9 | 8 | a1i | ⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ 𝐶 ) |
10 | 6 9 | ssnct | ⊢ ( 𝜑 → ¬ 𝐶 ≼ ω ) |