ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-pre-ltwlin GIF version

Axiom ax-pre-ltwlin 7880
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7838. (Contributed by Jim Kingdon, 12-Jan-2020.)
Assertion
Ref Expression
ax-pre-ltwlin ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))

Detailed syntax breakdown of Axiom ax-pre-ltwlin
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 7766 . . . 4 class
31, 2wcel 2141 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2141 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 2141 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 973 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cltrr 7771 . . . 4 class <
101, 4, 9wbr 3987 . . 3 wff 𝐴 < 𝐵
111, 6, 9wbr 3987 . . . 4 wff 𝐴 < 𝐶
126, 4, 9wbr 3987 . . . 4 wff 𝐶 < 𝐵
1311, 12wo 703 . . 3 wff (𝐴 < 𝐶𝐶 < 𝐵)
1410, 13wi 4 . 2 wff (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵))
158, 14wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff set class
This axiom is referenced by:  axltwlin  7980
  Copyright terms: Public domain W3C validator