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Theorem ordtri3OLD 5798
Description: Obsolete proof of ordtri3 5797 as of 24-Sep-2021. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ordtri3OLD ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))

Proof of Theorem ordtri3OLD
StepHypRef Expression
1 ordirr 5779 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq2 2719 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
32notbid 307 . . . . . 6 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐴𝐵))
41, 3syl5ib 234 . . . . 5 (𝐴 = 𝐵 → (Ord 𝐴 → ¬ 𝐴𝐵))
5 ordirr 5779 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
6 eleq2 2719 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
76notbid 307 . . . . . 6 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
85, 7syl5ibr 236 . . . . 5 (𝐴 = 𝐵 → (Ord 𝐵 → ¬ 𝐵𝐴))
94, 8anim12d 585 . . . 4 (𝐴 = 𝐵 → ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
109com12 32 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
11 pm4.56 515 . . 3 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
1210, 11syl6ib 241 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ (𝐴𝐵𝐵𝐴)))
13 ordtri3or 5793 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
14 df-3or 1055 . . . . 5 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
1513, 14sylib 208 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
16 or32 548 . . . 4 (((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1715, 16sylib 208 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1817ord 391 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵))
1912, 18impbid 202 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1053   = wceq 1523  wcel 2030  Ord word 5760
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-3or 1055  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-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by: (None)
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