Theorem xrlenltd

Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrlenltd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrlenltd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
Assertion	xrlenltd	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xrlenltd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrlenltd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrlenlt	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )