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Theorem ordtri3 6225
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 6207 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
21adantl 482 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵𝐵)
3 eleq2 2906 . . . . . 6 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
43notbid 319 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
52, 4syl5ibrcom 248 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
65pm4.71d 562 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴)))
7 pm5.61 996 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵𝐴))
8 pm4.52 980 . . . 4 (((𝐴 = 𝐵𝐵𝐴) ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
97, 8bitr3i 278 . . 3 ((𝐴 = 𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
106, 9syl6bb 288 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
11 ordtri2 6224 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
1211orbi1d 912 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1312notbid 319 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1410, 13bitr4d 283 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  Ord word 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192
This theorem is referenced by:  ordunisuc2  7547  tz7.48lem  8068  oacan  8164  omcan  8185  oecan  8205  omsmo  8271  omopthi  8274  inf3lem6  9085  cantnfp1lem3  9132  infpssrlem5  9718  fin23lem24  9733  isf32lem4  9767  om2uzf1oi  13311
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