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Axiom ax-pre-ltwlin 6756
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6727. (Contributed by Jim Kingdon, 12-Jan-2020.)
Assertion
Ref Expression
ax-pre-ltwlin ((A B 𝐶 ℝ) → (A < B → (A < 𝐶 𝐶 < B)))

Detailed syntax breakdown of Axiom ax-pre-ltwlin
StepHypRef Expression
1 cA . . . 4 class A
2 cr 6670 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cC . . . 4 class 𝐶
76, 2wcel 1390 . . 3 wff 𝐶
83, 5, 7w3a 884 . 2 wff (A B 𝐶 ℝ)
9 cltrr 6675 . . . 4 class <
101, 4, 9wbr 3755 . . 3 wff A < B
111, 6, 9wbr 3755 . . . 4 wff A < 𝐶
126, 4, 9wbr 3755 . . . 4 wff 𝐶 < B
1311, 12wo 628 . . 3 wff (A < 𝐶 𝐶 < B)
1410, 13wi 4 . 2 wff (A < B → (A < 𝐶 𝐶 < B))
158, 14wi 4 1 wff ((A B 𝐶 ℝ) → (A < B → (A < 𝐶 𝐶 < B)))
Colors of variables: wff set class
This axiom is referenced by:  axltwlin  6844
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