Theorem ltnelicc

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

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

Proof

Step	Hyp	Ref	Expression
1		ltnelicc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnelicc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		ltnelicc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		ltnelicc.clta	⊢ ( 𝜑 → 𝐶 < 𝐴 )
5	1	rexrd	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
6		xrltnle	⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶 ) )
7	3 5 6	syl2anc	⊢ ( 𝜑 → ( 𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶 ) )
8	4 7	mpbid	⊢ ( 𝜑 → ¬ 𝐴 ≤ 𝐶 )
9	8	intnanrd	⊢ ( 𝜑 → ¬ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
10		elicc4	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
11	5 2 3 10	syl3anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
12	9 11	mtbird	⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )