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Theorem axlttrn 6845
Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6757 with ordering on the extended reals. New proofs should use lttr 6849 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axlttrn ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))

Proof of Theorem axlttrn
StepHypRef Expression
1 ax-pre-lttrn 6757 . 2 ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
2 ltxrlt 6842 . . . 4 ((A B ℝ) → (A < BA < B))
323adant3 923 . . 3 ((A B 𝐶 ℝ) → (A < BA < B))
4 ltxrlt 6842 . . . 4 ((B 𝐶 ℝ) → (B < 𝐶B < 𝐶))
543adant1 921 . . 3 ((A B 𝐶 ℝ) → (B < 𝐶B < 𝐶))
63, 5anbi12d 442 . 2 ((A B 𝐶 ℝ) → ((A < B B < 𝐶) ↔ (A < B B < 𝐶)))
7 ltxrlt 6842 . . 3 ((A 𝐶 ℝ) → (A < 𝐶A < 𝐶))
873adant2 922 . 2 ((A B 𝐶 ℝ) → (A < 𝐶A < 𝐶))
91, 6, 83imtr4d 192 1 ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   wcel 1390   class class class wbr 3755  cr 6670   < cltrr 6675   < clt 6817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6734  ax-resscn 6735  ax-pre-lttrn 6757
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6819  df-mnf 6820  df-ltxr 6822
This theorem is referenced by:  lttr  6849
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