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Metamath Proof Explorer

Theorem lttri5d

Description: Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses lttri5d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
lttri5d.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
lttri5d.aneb ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
lttri5d.nlt ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )
Assertion lttri5d ( 𝜑 → 𝐴 < 𝐵 )

Proof

Step Hyp Ref Expression
1 lttri5d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 lttri5d.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 lttri5d.aneb ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
4 lttri5d.nlt ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )
5 1 rexrd ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
6 2 rexrd ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
7 5 6 3 4 xrlttri5d ⊢ ( 𝜑 → 𝐴 < 𝐵 )