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Metamath Proof Explorer

Theorem ltletr

Description: Transitive law. (Contributed by NM, 25-Aug-1999)

Ref Expression
Assertion ltletr ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) )

Proof

Step Hyp Ref Expression
1 leloe ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
2 1 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
3 lttr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
4 3 expcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
5 breq2 ⊢ ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 ↔ 𝐴 < 𝐶 ) )
6 5 biimpd ⊢ ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) )
7 6 a1i ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
8 4 7 jaod ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
9 2 8 sylbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ≤ 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
10 9 impcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) )