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Theorem poltletr 6090
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 6089 . . . . 5 (𝐶𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
213ad2ant3 1136 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
32adantl 483 . . 3 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
43anbi2d 630 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶))))
5 potr 5562 . . . . 5 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
65com12 32 . . . 4 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
7 breq2 5113 . . . . . 6 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
87biimpac 480 . . . . 5 ((𝐴𝑅𝐵𝐵 = 𝐶) → 𝐴𝑅𝐶)
98a1d 25 . . . 4 ((𝐴𝑅𝐵𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
106, 9jaodan 957 . . 3 ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
1110com12 32 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → 𝐴𝑅𝐶))
124, 11sylbid 239 1 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  cun 3912   class class class wbr 5109   I cid 5534   Po wpo 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-po 5549  df-xp 5643  df-rel 5644
This theorem is referenced by:  soltmin  6094
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