Axiom ax-pre-ltwlin 8293

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 8251. (Contributed by Jim Kingdon, 12-Jan-2020.)

Assertion

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

Detailed syntax breakdown of Axiom ax-pre-ltwlin

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 8179	. . . 4 class ℝ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2209	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 1009	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cltrr 8184	. . . 4 class <ℝ
10	1, 4, 9	wbr 4130	. . 3 wff 𝐴 <ℝ 𝐵
11	1, 6, 9	wbr 4130	. . . 4 wff 𝐴 <ℝ 𝐶
12	6, 4, 9	wbr 4130	. . . 4 wff 𝐶 <ℝ 𝐵
13	11, 12	wo 720	. . 3 wff (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)
14	10, 13	wi 4	. 2 wff (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

This axiom is used by:  axltwlin  8394