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Theorem ordtri3 6206
 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 6188 . . . . . 6 (Ord 𝐵 → ¬ 𝐵𝐵)
21adantl 486 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵𝐵)
3 eleq2 2841 . . . . . 6 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
43notbid 322 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐵))
52, 4syl5ibrcom 250 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
65pm4.71d 566 . . 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 280 . . 3 ((𝐴 = 𝐵 ∧ ¬ 𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴))
106, 9bitrdi 290 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
11 ordtri2 6205 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
1211orbi1d 915 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1312notbid 322 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵𝐵𝐴) ↔ ¬ (¬ (𝐴 = 𝐵𝐵𝐴) ∨ 𝐵𝐴)))
1410, 13bitr4d 285 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  Ord word 6169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-ord 6173 This theorem is referenced by:  ordunisuc2  7559  tz7.48lem  8088  oacan  8185  omcan  8206  oecan  8226  omsmo  8292  omopthi  8295  inf3lem6  9122  cantnfp1lem3  9169  infpssrlem5  9760  fin23lem24  9775  isf32lem4  9809  om2uzf1oi  13363
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