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Metamath Proof Explorer

Theorem lelttrd

Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006)

Ref Expression
Hypotheses ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
letrd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
lelttrd.4 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
lelttrd.5 ⊢ ( 𝜑 → 𝐵 < 𝐶 )
Assertion lelttrd ( 𝜑 → 𝐴 < 𝐶 )

Proof

Step Hyp Ref Expression
1 ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 letrd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4 lelttrd.4 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5 lelttrd.5 ⊢ ( 𝜑 → 𝐵 < 𝐶 )
6 lelttr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
7 1 2 3 6 syl3anc ⊢ ( 𝜑 → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
8 4 5 7 mp2and ⊢ ( 𝜑 → 𝐴 < 𝐶 )