Axiom ax-pre-lttrn 10946

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 10922. Note: The more general version for extended reals is axlttrn 11047. Normally new proofs would use lttr 11051. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-lttrn	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Detailed syntax breakdown of Axiom ax-pre-lttrn

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 10870	. . . 4 class ℝ
3	1, 2	wcel 2106	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2106	. . 3 wff 𝐵 ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2106	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cltrr 10875	. . . . 5 class <ℝ
10	1, 4, 9	wbr 5074	. . . 4 wff 𝐴 <ℝ 𝐵
11	4, 6, 9	wbr 5074	. . . 4 wff 𝐵 <ℝ 𝐶
12	10, 11	wa 396	. . 3 wff (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶)
13	1, 6, 9	wbr 5074	. . 3 wff 𝐴 <ℝ 𝐶
14	12, 13	wi 4	. 2 wff ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

This axiom is referenced by:  axlttrn  11047