Theorem gtnelioc

Description: A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	gtnelioc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	gtnelioc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	gtnelioc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
	gtnelioc.bltc	⊢ ( 𝜑 → 𝐵 < 𝐶 )
Assertion	gtnelioc	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gtnelioc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		gtnelioc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		gtnelioc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		gtnelioc.bltc	⊢ ( 𝜑 → 𝐵 < 𝐶 )
5	2	rexrd	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
6		xrltnle	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵 ) )
7	5 3 6	syl2anc	⊢ ( 𝜑 → ( 𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵 ) )
8	4 7	mpbid	⊢ ( 𝜑 → ¬ 𝐶 ≤ 𝐵 )
9	8	intn3an3d	⊢ ( 𝜑 → ¬ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
10		elioc2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
11	1 2 10	syl2anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
12	9 11	mtbird	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )