This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )