Theorem lenelioc

Description: A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021)

	Ref	Expression
Hypotheses	lenelioc.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	lenelioc.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	lenelioc.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
	lenelioc.4	⊢ ( 𝜑 → 𝐶 ≤ 𝐴 )
Assertion	lenelioc	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lenelioc.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		lenelioc.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		lenelioc.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		lenelioc.4	⊢ ( 𝜑 → 𝐶 ≤ 𝐴 )
5	3 1	xrlenltd	⊢ ( 𝜑 → ( 𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶 ) )
6	4 5	mpbid	⊢ ( 𝜑 → ¬ 𝐴 < 𝐶 )
7	6	intn3an2d	⊢ ( 𝜑 → ¬ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
8		elioc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
9	1 2 8	syl2anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
10	7 9	mtbird	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )