Theorem gtnelicc

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

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

Proof

Step	Hyp	Ref	Expression
1		gtnelicc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		gtnelicc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		gtnelicc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		gtnelicc.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	intnand	⊢ ( 𝜑 → ¬ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
10		elicc4	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
11	1 5 3 10	syl3anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
12	9 11	mtbird	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )