Theorem lelttr

Description: Transitive law. (Contributed by NM, 23-May-1999)

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

Proof

Step	Hyp	Ref	Expression
1		leloe	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
2	1	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3		lttr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
4	3	expd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 → ( 𝐵 < 𝐶 → 𝐴 < 𝐶 ) ) )
5		breq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 < 𝐶 ↔ 𝐵 < 𝐶 ) )
6	5	biimprd	⊢ ( 𝐴 = 𝐵 → ( 𝐵 < 𝐶 → 𝐴 < 𝐶 ) )
7	6	a1i	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 = 𝐵 → ( 𝐵 < 𝐶 → 𝐴 < 𝐶 ) ) )
8	4 7	jaod	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) → ( 𝐵 < 𝐶 → 𝐴 < 𝐶 ) ) )
9	2 8	sylbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → ( 𝐵 < 𝐶 → 𝐴 < 𝐶 ) ) )
10	9	impd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )