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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