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Theorem ordtr3OLD 5729
Description: Obsolete proof of ordtr3 5728 as of 24-Sep-2021. (Contributed by Mario Carneiro, 30-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ordtr3OLD ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))

Proof of Theorem ordtr3OLD
StepHypRef Expression
1 simpr 477 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → Ord 𝐶)
2 ordelord 5704 . . . . . 6 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
32adantlr 750 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → Ord 𝐴)
4 ordtri1 5715 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐴) → (𝐶𝐴 ↔ ¬ 𝐴𝐶))
51, 3, 4syl2an2r 875 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐶𝐴 ↔ ¬ 𝐴𝐶))
6 ordtr2 5727 . . . . . . 7 ((Ord 𝐶 ∧ Ord 𝐵) → ((𝐶𝐴𝐴𝐵) → 𝐶𝐵))
76ancoms 469 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → ((𝐶𝐴𝐴𝐵) → 𝐶𝐵))
87expcomd 454 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐶𝐴𝐶𝐵)))
98imp 445 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐶𝐴𝐶𝐵))
105, 9sylbird 250 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
1110orrd 393 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
1211ex 450 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wcel 1987  wss 3555  Ord word 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685
This theorem is referenced by: (None)
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