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Theorem axlttrn 8061
Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7960 with ordering on the extended reals. New proofs should use lttr 8066 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axlttrn ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Proof of Theorem axlttrn
StepHypRef Expression
1 ax-pre-lttrn 7960 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
2 ltxrlt 8058 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
323adant3 1019 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
4 ltxrlt 8058 . . . 4 ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶𝐵 < 𝐶))
543adant1 1017 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶𝐵 < 𝐶))
63, 5anbi12d 473 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
7 ltxrlt 8058 . . 3 ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶𝐴 < 𝐶))
873adant2 1018 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶𝐴 < 𝐶))
91, 6, 83imtr4d 203 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wcel 2160   class class class wbr 4021  cr 7845   < cltrr 7850   < clt 8027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-pre-lttrn 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-pnf 8029  df-mnf 8030  df-ltxr 8032
This theorem is referenced by:  lttr  8066
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