Theorem leltletr

Description: Transitive law, weaker form of lelttr . (Contributed by AV, 14-Oct-2018)

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

Proof

Step	Hyp	Ref	Expression
1		3simpb	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) )
2		lelttr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
3		ltle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) )
4	1 2 3	sylsyld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 ≤ 𝐶 ) )