Metamath Proof Explorer


Theorem iocnct

Description: A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021)

Ref Expression
Hypotheses iocnct.a ( 𝜑𝐴 ∈ ℝ* )
iocnct.b ( 𝜑𝐵 ∈ ℝ* )
iocnct.l ( 𝜑𝐴 < 𝐵 )
iocnct.c 𝐶 = ( 𝐴 (,] 𝐵 )
Assertion iocnct ( 𝜑 → ¬ 𝐶 ≼ ω )

Proof

Step Hyp Ref Expression
1 iocnct.a ( 𝜑𝐴 ∈ ℝ* )
2 iocnct.b ( 𝜑𝐵 ∈ ℝ* )
3 iocnct.l ( 𝜑𝐴 < 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 )