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Axiom ax-pre-ltwlin 6969
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6929. (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 6860 . . . 4 class
31, 2wcel 1393 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1393 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 1393 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 885 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cltrr 6865 . . . 4 class <
101, 4, 9wbr 3761 . . 3 wff 𝐴 < 𝐵
111, 6, 9wbr 3761 . . . 4 wff 𝐴 < 𝐶
126, 4, 9wbr 3761 . . . 4 wff 𝐶 < 𝐵
1311, 12wo 629 . . 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  7058
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