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Metamath Proof Explorer

Theorem ltletri

Description: 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt.1 ⊢ 𝐴 ∈ ℝ
lt.2 ⊢ 𝐵 ∈ ℝ
lt.3 ⊢ 𝐶 ∈ ℝ
Assertion ltletri ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 )

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ 𝐴 ∈ ℝ
2 lt.2 ⊢ 𝐵 ∈ ℝ
3 lt.3 ⊢ 𝐶 ∈ ℝ
4 ltletr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) )
5 1 2 3 4 mp3an ⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 )