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Axiom ax-pre-lttrn 7727
 Description: Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7685. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-lttrn ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Detailed syntax breakdown of Axiom ax-pre-lttrn
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 7612 . . . 4 class
31, 2wcel 1480 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1480 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 1480 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 962 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cltrr 7617 . . . . 5 class <
101, 4, 9wbr 3924 . . . 4 wff 𝐴 < 𝐵
114, 6, 9wbr 3924 . . . 4 wff 𝐵 < 𝐶
1210, 11wa 103 . . 3 wff (𝐴 < 𝐵𝐵 < 𝐶)
131, 6, 9wbr 3924 . . 3 wff 𝐴 < 𝐶
1412, 13wi 4 . 2 wff ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)
158, 14wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 Colors of variables: wff set class This axiom is referenced by:  axlttrn  7826
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