Axiom ax-pre-lttri 10876

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 10852. Note: The more general version for extended reals is axlttri 10977. Normally new proofs would use xrlttri 12802. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-lttri	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Detailed syntax breakdown of Axiom ax-pre-lttri

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 10801	. . . 4 class ℝ
3	1, 2	wcel 2108	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2108	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cltrr 10806	. . . 4 class <ℝ
8	1, 4, 7	wbr 5070	. . 3 wff 𝐴 <ℝ 𝐵
9	1, 4	wceq 1539	. . . . 5 wff 𝐴 = 𝐵
10	4, 1, 7	wbr 5070	. . . . 5 wff 𝐵 <ℝ 𝐴
11	9, 10	wo 843	. . . 4 wff (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)
12	11	wn 3	. . 3 wff ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)
13	8, 12	wb 205	. 2 wff (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))
14	6, 13	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))

This axiom is referenced by:  axlttri  10977