Theorem xrgtnelicc

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021)

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

Proof

Step	Hyp	Ref	Expression
1		xrgtnelicc.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrgtnelicc.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrgtnelicc.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		xrgtnelicc.4	⊢ ( 𝜑 → 𝐵 < 𝐶 )
5		xrltnle	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵 ) )
6	2 3 5	syl2anc	⊢ ( 𝜑 → ( 𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵 ) )
7	4 6	mpbid	⊢ ( 𝜑 → ¬ 𝐶 ≤ 𝐵 )
8	7	intnand	⊢ ( 𝜑 → ¬ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
9		elicc4	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
10	1 2 3 9	syl3anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
11	8 10	mtbird	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )