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Theorem poltletr 5563
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 5562 . . . . 5 (𝐶𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
213ad2ant3 1104 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
32adantl 481 . . 3 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
43anbi2d 740 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶))))
5 potr 5076 . . . . 5 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
65com12 32 . . . 4 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
7 breq2 4689 . . . . . 6 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
87biimpac 502 . . . . 5 ((𝐴𝑅𝐵𝐵 = 𝐶) → 𝐴𝑅𝐶)
98a1d 25 . . . 4 ((𝐴𝑅𝐵𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
106, 9jaodan 843 . . 3 ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
1110com12 32 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → 𝐴𝑅𝐶))
124, 11sylbid 230 1 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  cun 3605   class class class wbr 4685   I cid 5052   Po wpo 5062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-po 5064  df-xp 5149  df-rel 5150
This theorem is referenced by:  soltmin  5567
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