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Theorem ordtri4 6211
 Description: A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))

Proof of Theorem ordtri4
StepHypRef Expression
1 eqss 3909 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
2 ordtri1 6207 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
32ancoms 462 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
43anbi2d 631 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
51, 4syl5bb 286 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  Ord word 6173 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177 This theorem is referenced by:  carduni  9456  alephfp  9581  newbday  33674
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