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Theorem ordtri3OLD 5677
 Description: Obsolete proof of ordtri3 5676 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 5658 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq2 2677 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
32notbid 307 . . . . . 6 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐴𝐵))
41, 3syl5ib 233 . . . . 5 (𝐴 = 𝐵 → (Ord 𝐴 → ¬ 𝐴𝐵))
5 ordirr 5658 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
6 eleq2 2677 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
76notbid 307 . . . . . 6 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
85, 7syl5ibr 235 . . . . 5 (𝐴 = 𝐵 → (Ord 𝐵 → ¬ 𝐵𝐴))
94, 8anim12d 584 . . . 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 240 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ (𝐴𝐵𝐵𝐴)))
13 ordtri3or 5672 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
14 df-3or 1032 . . . . 5 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
1513, 14sylib 207 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
16 or32 548 . . . 4 (((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1715, 16sylib 207 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1817ord 391 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵))
1912, 18impbid 201 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  Ord word 5639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by: (None)
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