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Theorem poltletr 4909
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 4908 . . . . 5 (𝐶𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
213ad2ant3 989 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
32adantl 275 . . 3 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
43anbi2d 459 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶))))
5 potr 4200 . . . . 5 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
65com12 30 . . . 4 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
7 breq2 3903 . . . . . 6 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
87biimpac 296 . . . . 5 ((𝐴𝑅𝐵𝐵 = 𝐶) → 𝐴𝑅𝐶)
98a1d 22 . . . 4 ((𝐴𝑅𝐵𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
106, 9jaodan 771 . . 3 ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
1110com12 30 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → 𝐴𝑅𝐶))
124, 11sylbid 149 1 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 682  w3a 947   = wceq 1316  wcel 1465  cun 3039   class class class wbr 3899   I cid 4180   Po wpo 4186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-xp 4515  df-rel 4516
This theorem is referenced by: (None)
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