Theorem relin01

Description: An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013)

	Ref	Expression
Assertion	relin01	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∨ 1 ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		letric	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐴 ≤ 1 ∨ 1 ≤ 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 1 ∨ 1 ≤ 𝐴 ) )
4		0re	⊢ 0 ∈ ℝ
5		letric	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 ≤ 0 ∨ 0 ≤ 𝐴 ) )
6	4 5	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 0 ∨ 0 ≤ 𝐴 ) )
7		pm3.21	⊢ ( 𝐴 ≤ 1 → ( 0 ≤ 𝐴 → ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) )
8	7	orim2d	⊢ ( 𝐴 ≤ 1 → ( ( 𝐴 ≤ 0 ∨ 0 ≤ 𝐴 ) → ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) )
9	6 8	syl5com	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 1 → ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) )
10	9	orim1d	⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 ≤ 1 ∨ 1 ≤ 𝐴 ) → ( ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ∨ 1 ≤ 𝐴 ) ) )
11	3 10	mpd	⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ∨ 1 ≤ 𝐴 ) )
12		df-3or	⊢ ( ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∨ 1 ≤ 𝐴 ) ↔ ( ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ∨ 1 ≤ 𝐴 ) )
13	11 12	sylibr	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 0 ∨ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∨ 1 ≤ 𝐴 ) )