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Theorem domtriord 8066
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 8040 . . . . 5 ((𝐵𝐴𝐴𝐵) → 𝐵𝐴)
21expcom 451 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐵𝐴))
32a1i 11 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐵𝐴𝐵𝐴)))
4 iman 440 . . . 4 ((𝐵𝐴𝐵𝐴) ↔ ¬ (𝐵𝐴 ∧ ¬ 𝐵𝐴))
5 brsdom 7938 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐴))
64, 5xchbinxr 325 . . 3 ((𝐵𝐴𝐵𝐴) ↔ ¬ 𝐵𝐴)
73, 6syl6ib 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ¬ 𝐵𝐴))
8 onelss 5735 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
9 ssdomg 7961 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
108, 9syld 47 . . . . . . . . 9 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
1110adantl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
1211con3d 148 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → ¬ 𝐴𝐵))
13 ontri1 5726 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1413ancoms 469 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1512, 14sylibrd 249 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
16 ssdomg 7961 . . . . . . 7 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
1716adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴𝐵𝐴))
1815, 17syld 47 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
19 ensym 7965 . . . . . . . 8 (𝐵𝐴𝐴𝐵)
20 endom 7942 . . . . . . . 8 (𝐴𝐵𝐴𝐵)
2119, 20syl 17 . . . . . . 7 (𝐵𝐴𝐴𝐵)
2221con3i 150 . . . . . 6 𝐴𝐵 → ¬ 𝐵𝐴)
2322a1i 11 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → ¬ 𝐵𝐴))
2418, 23jcad 555 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → (𝐵𝐴 ∧ ¬ 𝐵𝐴)))
2524, 5syl6ibr 242 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
2625con1d 139 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵𝐴𝐴𝐵))
277, 26impbid 202 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  wss 3560   class class class wbr 4623  Oncon0 5692  cen 7912  cdom 7913  csdm 7914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918
This theorem is referenced by:  sdomel  8067  cardsdomel  8760  alephord  8858  alephsucdom  8862  alephdom2  8870
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