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Axiom ax-pre-lttri 10298
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 10274. Note: The more general version for extended reals is axlttri 10397. Normally new proofs would use xrlttri 12191. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-lttri ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Detailed syntax breakdown of Axiom ax-pre-lttri
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 10223 . . . 4 class
31, 2wcel 2157 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2157 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 384 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cltrr 10228 . . . 4 class <
81, 4, 7wbr 4851 . . 3 wff 𝐴 < 𝐵
91, 4wceq 1637 . . . . 5 wff 𝐴 = 𝐵
104, 1, 7wbr 4851 . . . . 5 wff 𝐵 < 𝐴
119, 10wo 865 . . . 4 wff (𝐴 = 𝐵𝐵 < 𝐴)
1211wn 3 . . 3 wff ¬ (𝐴 = 𝐵𝐵 < 𝐴)
138, 12wb 197 . 2 wff (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴))
146, 13wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
This axiom is referenced by:  axlttri  10397
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