Axiom ax-pre-ltwlin 6796

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6767. (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 6710	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1390	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 884	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cltrr 6715	. . . 4 class <ℝ
10	1, 4, 9	wbr 3755	. . 3 wff A <ℝ B
11	1, 6, 9	wbr 3755	. . . 4 wff A <ℝ 𝐶
12	6, 4, 9	wbr 3755	. . . 4 wff 𝐶 <ℝ B
13	11, 12	wo 628	. . 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 referenced by:  axltwlin  6884