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Metamath Proof Explorer

Theorem icossico2d

Description: Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	icossico2d.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
	icossico2d.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
	icossico2d.3	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
Assertion	icossico2d	⊢ ( 𝜑 → ( 𝐴 [,) 𝐶 ) ⊆ ( 𝐵 [,) 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		icossico2d.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
2		icossico2d.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
3		icossico2d.3	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
4	2	xrleidd	⊢ ( 𝜑 → 𝐶 ≤ 𝐶 )
5		icossico	⊢ ( ( ( 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐶 ≤ 𝐶 ) ) → ( 𝐴 [,) 𝐶 ) ⊆ ( 𝐵 [,) 𝐶 ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( 𝐴 [,) 𝐶 ) ⊆ ( 𝐵 [,) 𝐶 ) )