Theorem axlttrn

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn with ordering on the extended reals. New proofs should use lttr instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

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

Proof

Step	Hyp	Ref	Expression
1		ax-pre-lttrn	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶 ) → 𝐴 <ℝ 𝐶 ) )
2		ltxrlt	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵 ) )
3	2	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵 ) )
4		ltxrlt	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 ↔ 𝐵 <ℝ 𝐶 ) )
5	4	3adant1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 ↔ 𝐵 <ℝ 𝐶 ) )
6	3 5	anbi12d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ↔ ( 𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶 ) ) )
7		ltxrlt	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶 ) )
8	7	3adant2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶 ) )
9	1 6 8	3imtr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )