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Metamath Proof Explorer

Theorem xrltnled

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses xrltnled.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
xrltnled.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
Assertion xrltnled ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )

Proof

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