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Theorem poltletr 4750
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 4749 . . . . 5 (𝐶𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
213ad2ant3 936 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
32adantl 266 . . 3 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶𝐵 = 𝐶)))
43anbi2d 445 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶))))
5 potr 4070 . . . . 5 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
65com12 30 . . . 4 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
7 breq2 3793 . . . . . 6 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
87biimpac 286 . . . . 5 ((𝐴𝑅𝐵𝐵 = 𝐶) → 𝐴𝑅𝐶)
98a1d 22 . . . 4 ((𝐴𝑅𝐵𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
106, 9jaodan 719 . . 3 ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑅𝐶))
1110com12 30 . 2 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶𝐵 = 𝐶)) → 𝐴𝑅𝐶))
124, 11sylbid 143 1 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 637  w3a 894   = wceq 1257  wcel 1407  cun 2940   class class class wbr 3789   I cid 4050   Po wpo 4056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-id 4055  df-po 4058  df-xp 4376  df-rel 4377
This theorem is referenced by: (None)
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