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Theorem poltletr 4725
Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
poltletr ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))

Proof of Theorem poltletr
StepHypRef Expression
1 poleloe 4724 . . . . 5 (𝐶𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
213ad2ant3 927 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
32adantl 262 . . 3 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
43anbi2d 437 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶))))
5 potr 4045 . . . . 5 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
65com12 27 . . . 4 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
7 breq2 3768 . . . . . 6 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
87biimpac 282 . . . . 5 ((𝐴𝑅𝐵𝐵 = 𝐶) → 𝐴𝑅𝐶)
98a1d 22 . . . 4 ((𝐴𝑅𝐵𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
106, 9jaodan 710 . . 3 ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
1110com12 27 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → 𝐴𝑅𝐶))
124, 11sylbid 139 1 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629  w3a 885   = wceq 1243  wcel 1393  cun 2915   class class class wbr 3764   I cid 4025   Po wpo 4031
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-xp 4351  df-rel 4352
This theorem is referenced by: (None)
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