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Theorem ordtri3 5723
Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
ordtri3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))

Proof of Theorem ordtri3
StepHypRef Expression
1 ordirr 5705 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
21adantl 482 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵𝐵)
3 eleq2 2687 . . . . . 6 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
43notbid 308 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
52, 4syl5ibrcom 237 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
65pm4.71d 665 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴)))
7 pm5.61 748 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴))
8 pm4.52 512 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
97, 8bitr3i 266 . . 3 ((𝐴 = 𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
106, 9syl6bb 276 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
11 ordtri2 5722 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
1211orbi1d 738 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1312notbid 308 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1410, 13bitr4d 271 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   = wceq 1480  wcel 1987  Ord word 5686
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 4746  ax-nul 4754  ax-pr 4872
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 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690
This theorem is referenced by:  ordunisuc2  6998  tz7.48lem  7488  oacan  7580  omcan  7601  oecan  7621  omsmo  7686  omopthi  7689  inf3lem6  8481  cantnfp1lem3  8528  infpssrlem5  9080  fin23lem24  9095  isf32lem4  9129  om2uzf1oi  12699  nodenselem4  31574
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