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Theorem domtriord 8652
Description: Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
Assertion
Ref Expression
domtriord ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Proof of Theorem domtriord
StepHypRef Expression
1 sbth 8626 . . . . 5 ((𝐵𝐴𝐴𝐵) → 𝐵𝐴)
21expcom 414 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐵𝐴))
32a1i 11 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐵𝐴𝐵𝐴)))
4 iman 402 . . . 4 ((𝐵𝐴𝐵𝐴) ↔ ¬ (𝐵𝐴 ∧ ¬ 𝐵𝐴))
5 brsdom 8521 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐴))
64, 5xchbinxr 336 . . 3 ((𝐵𝐴𝐵𝐴) ↔ ¬ 𝐵𝐴)
73, 6syl6ib 252 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ¬ 𝐵𝐴))
8 onelss 6227 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
9 ssdomg 8544 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
108, 9syld 47 . . . . . . . . 9 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
1110adantl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
1211con3d 155 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → ¬ 𝐴𝐵))
13 ontri1 6219 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1413ancoms 459 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1512, 14sylibrd 260 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
16 ssdomg 8544 . . . . . . 7 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
1716adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴𝐵𝐴))
1815, 17syld 47 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
19 ensym 8547 . . . . . . 7 (𝐵𝐴𝐴𝐵)
20 endom 8525 . . . . . . 7 (𝐴𝐵𝐴𝐵)
2119, 20syl 17 . . . . . 6 (𝐵𝐴𝐴𝐵)
2221con3i 157 . . . . 5 𝐴𝐵 → ¬ 𝐵𝐴)
2318, 22jca2 514 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → (𝐵𝐴 ∧ ¬ 𝐵𝐴)))
2423, 5syl6ibr 253 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
2524con1d 147 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵𝐴𝐴𝐵))
267, 25impbid 213 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2105  wss 3935   class class class wbr 5058  Oncon0 6185  cen 8495  cdom 8496  csdm 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501
This theorem is referenced by:  sdomel  8653  cardsdomel  9392  alephord  9490  alephsucdom  9494  alephdom2  9502
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