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Theorem ordtri3 6399
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 6381 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
21adantl 480 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵𝐵)
3 eleq2 2820 . . . . . 6 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
43notbid 317 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
52, 4syl5ibrcom 246 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
65pm4.71d 560 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴)))
7 pm5.61 997 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴))
8 pm4.52 981 . . . 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 6398 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
1211orbi1d 913 . . 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 394  wo 843   = wceq 1539  wcel 2104  Ord word 6362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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 6366
This theorem is referenced by:  ordunisuc2  7835  tz7.48lem  8443  oacan  8550  omcan  8571  oecan  8591  omsmo  8659  omopthi  8662  inf3lem6  9630  cantnfp1lem3  9677  infpssrlem5  10304  fin23lem24  10319  isf32lem4  10353  om2uzf1oi  13922  ordnexbtwnsuc  42319
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