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Metamath Proof Explorer

Theorem iccss

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 20-Feb-2015)

		Ref	Expression
	Assertion	iccss	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rexr	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ^* )
2		rexr	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ^* )
3	1 2	anim12i	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) )
4		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
5		xrletr	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^* ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝑤 ) → 𝐴 ≤ 𝑤 ) )
6		xrletr	⊢ ( ( 𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( ( 𝑤 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
7	4 4 5 6	ixxss12	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )
8	3 7	sylan	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )