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Theorem ordtri3 6397
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 6379 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
21adantl 482 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵𝐵)
3 eleq2 2822 . . . . . 6 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
43notbid 317 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
52, 4syl5ibrcom 246 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
65pm4.71d 562 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴)))
7 pm5.61 999 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴))
8 pm4.52 983 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
97, 8bitr3i 276 . . 3 ((𝐴 = 𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
106, 9bitrdi 286 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
11 ordtri2 6396 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
1211orbi1d 915 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1312notbid 317 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1410, 13bitr4d 281 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  Ord word 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364
This theorem is referenced by:  ordunisuc2  7829  tz7.48lem  8437  oacan  8544  omcan  8565  oecan  8585  omsmo  8653  omopthi  8656  inf3lem6  9624  cantnfp1lem3  9671  infpssrlem5  10298  fin23lem24  10313  isf32lem4  10347  om2uzf1oi  13914  ordnexbtwnsuc  42002
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