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Theorem ordtr3OLD 5883
Description: Obsolete proof of ordtr3 5882 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 479 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → Ord 𝐶)
2 ordelord 5858 . . . . . 6 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
32adantlr 753 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → Ord 𝐴)
4 ordtri1 5869 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐴) → (𝐶𝐴 ↔ ¬ 𝐴𝐶))
51, 3, 4syl2an2r 911 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐶𝐴 ↔ ¬ 𝐴𝐶))
6 ordtr2 5881 . . . . . . 7 ((Ord 𝐶 ∧ Ord 𝐵) → ((𝐶𝐴𝐴𝐵) → 𝐶𝐵))
76ancoms 468 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → ((𝐶𝐴𝐴𝐵) → 𝐶𝐵))
87expcomd 453 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐶𝐴𝐶𝐵)))
98imp 444 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐶𝐴𝐶𝐵))
105, 9sylbird 250 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
1110orrd 392 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
1211ex 449 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wcel 2103  wss 3680  Ord word 5835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839
This theorem is referenced by: (None)
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