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Theorem ordtri3OLD 5719
Description: Obsolete proof of ordtri3 5718 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 5700 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq2 2687 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
32notbid 308 . . . . . 6 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐴𝐵))
41, 3syl5ib 234 . . . . 5 (𝐴 = 𝐵 → (Ord 𝐴 → ¬ 𝐴𝐵))
5 ordirr 5700 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
6 eleq2 2687 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
76notbid 308 . . . . . 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 pm4.56 516 . . 3 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
1210, 11syl6ib 241 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ (𝐴𝐵𝐵𝐴)))
13 ordtri3or 5714 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
14 df-3or 1037 . . . . 5 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
1513, 14sylib 208 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
16 or32 549 . . . 4 (((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1715, 16sylib 208 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
1817ord 392 . 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 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  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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