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Axiom ax-pre-lttrn 9867
Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9843. Note: The more general version for extended reals is axlttrn 9961. Normally new proofs would use lttr 9965. (New usage is discouraged.) (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 9791 . . . 4 class
31, 2wcel 1976 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1976 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 1976 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 1030 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cltrr 9796 . . . . 5 class <
101, 4, 9wbr 4577 . . . 4 wff 𝐴 < 𝐵
114, 6, 9wbr 4577 . . . 4 wff 𝐵 < 𝐶
1210, 11wa 382 . . 3 wff (𝐴 < 𝐵𝐵 < 𝐶)
131, 6, 9wbr 4577 . . 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 setvar class
This axiom is referenced by:  axlttrn  9961
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