Theorem xrleid

Description: 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007)

	Ref	Expression
Assertion	xrleid	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2	1	olci	⊢ ( 𝐴 < 𝐴 ∨ 𝐴 = 𝐴 )
3		xrleloe	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐴 ≤ 𝐴 ↔ ( 𝐴 < 𝐴 ∨ 𝐴 = 𝐴 ) ) )
4	2 3	mpbiri	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → 𝐴 ≤ 𝐴 )
5	4	anidms	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )