Metamath Proof Explorer

Theorem iccdisj

Description: If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024)

		Ref	Expression
	Assertion	iccdisj	⊢ ( ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ) ) ∧ 𝐵 < 𝐶 ) → ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐶 [,] 𝐷 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		simplll	⊢ ( ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ) ) ∧ 𝐵 < 𝐶 ) → 𝐴 ∈ ℝ^* )
2		simplrr	⊢ ( ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ) ) ∧ 𝐵 < 𝐶 ) → 𝐷 ∈ ℝ^* )
3		simpr	⊢ ( ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ) ) ∧ 𝐵 < 𝐶 ) → 𝐵 < 𝐶 )
4		iccdisj2	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐵 < 𝐶 ) → ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐶 [,] 𝐷 ) ) = ∅ )
5	1 2 3 4	syl3anc	⊢ ( ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ) ) ∧ 𝐵 < 𝐶 ) → ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐶 [,] 𝐷 ) ) = ∅ )