Theorem xrleloe

Description: 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006)

	Ref	Expression
Assertion	xrleloe	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrlenlt	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
2		xrlttri	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 < 𝐴 ↔ ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 < 𝐴 ↔ ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) )
4	3	con2bid	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ↔ ¬ 𝐵 < 𝐴 ) )
5		eqcom	⊢ ( 𝐵 = 𝐴 ↔ 𝐴 = 𝐵 )
6	5	orbi1i	⊢ ( ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 < 𝐵 ) )
7		orcom	⊢ ( ( 𝐴 = 𝐵 ∨ 𝐴 < 𝐵 ) ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
8	6 7	bitri	⊢ ( ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
9	4 8	bitr3di	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ 𝐵 < 𝐴 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
10	1 9	bitrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )