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Theorem ordtri2 6198
Description: A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
Assertion
Ref Expression
ordtri2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Proof of Theorem ordtri2
StepHypRef Expression
1 ordsseleq 6192 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
2 eqcom 2808 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
32orbi2i 910 . . . . . 6 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
4 orcom 867 . . . . . 6 ((𝐵𝐴𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐵𝐴))
53, 4bitri 278 . . . . 5 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐵𝐴))
61, 5syl6bb 290 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ (𝐴 = 𝐵𝐵𝐴)))
7 ordtri1 6196 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
86, 7bitr3d 284 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → ((𝐴 = 𝐵𝐵𝐴) ↔ ¬ 𝐴𝐵))
98ancoms 462 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 = 𝐵𝐵𝐴) ↔ ¬ 𝐴𝐵))
109con2bid 358 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wss 3884  Ord word 6162
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166
This theorem is referenced by:  ordtri3  6199  ord0eln0  6217  oaord  8160  omord2  8180  oeord  8201  nnaord  8232  nnmord  8245  noextenddif  33283
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