ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poltletr Structured version   GIF version

Theorem poltletr 4668
Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
poltletr ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B B(𝑅 ∪ I )𝐶) → A𝑅𝐶))

Proof of Theorem poltletr
StepHypRef Expression
1 poleloe 4667 . . . . 5 (𝐶 𝑋 → (B(𝑅 ∪ I )𝐶 ↔ (B𝑅𝐶 B = 𝐶)))
213ad2ant3 926 . . . 4 ((A 𝑋 B 𝑋 𝐶 𝑋) → (B(𝑅 ∪ I )𝐶 ↔ (B𝑅𝐶 B = 𝐶)))
32adantl 262 . . 3 ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → (B(𝑅 ∪ I )𝐶 ↔ (B𝑅𝐶 B = 𝐶)))
43anbi2d 437 . 2 ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B B(𝑅 ∪ I )𝐶) ↔ (A𝑅B (B𝑅𝐶 B = 𝐶))))
5 potr 4036 . . . . 5 ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))
65com12 27 . . . 4 ((A𝑅B B𝑅𝐶) → ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → A𝑅𝐶))
7 breq2 3759 . . . . . 6 (B = 𝐶 → (A𝑅BA𝑅𝐶))
87biimpac 282 . . . . 5 ((A𝑅B B = 𝐶) → A𝑅𝐶)
98a1d 22 . . . 4 ((A𝑅B B = 𝐶) → ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → A𝑅𝐶))
106, 9jaodan 709 . . 3 ((A𝑅B (B𝑅𝐶 B = 𝐶)) → ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → A𝑅𝐶))
1110com12 27 . 2 ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B (B𝑅𝐶 B = 𝐶)) → A𝑅𝐶))
124, 11sylbid 139 1 ((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B B(𝑅 ∪ I )𝐶) → A𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  cun 2909   class class class wbr 3755   I cid 4016   Po wpo 4022
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-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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-xp 4294  df-rel 4295
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator