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Theorem ordtri3or 6403
Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri3or ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Proof of Theorem ordtri3or
StepHypRef Expression
1 ordin 6401 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordirr 6389 . . . . . 6 (Ord (𝐴𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
31, 2syl 17 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
4 ianor 979 . . . . . 6 (¬ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
5 elin 3960 . . . . . . 7 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
6 incom 4199 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
76eleq1i 2816 . . . . . . . 8 ((𝐴𝐵) ∈ 𝐵 ↔ (𝐵𝐴) ∈ 𝐵)
87anbi2i 621 . . . . . . 7 (((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
95, 8bitri 274 . . . . . 6 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
104, 9xchnxbir 332 . . . . 5 (¬ (𝐴𝐵) ∈ (𝐴𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
113, 10sylib 217 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
12 inss1 4227 . . . . . . . . . 10 (𝐴𝐵) ⊆ 𝐴
13 ordsseleq 6400 . . . . . . . . . 10 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ⊆ 𝐴 ↔ ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴)))
1412, 13mpbii 232 . . . . . . . . 9 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
151, 14sylan 578 . . . . . . . 8 (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1615anabss1 664 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1716ord 862 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 → (𝐴𝐵) = 𝐴))
18 dfss2 3962 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
1917, 18imbitrrdi 251 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴𝐴𝐵))
20 ordin 6401 . . . . . . . . 9 ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵𝐴))
21 inss1 4227 . . . . . . . . . 10 (𝐵𝐴) ⊆ 𝐵
22 ordsseleq 6400 . . . . . . . . . 10 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ⊆ 𝐵 ↔ ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵)))
2321, 22mpbii 232 . . . . . . . . 9 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2420, 23sylan 578 . . . . . . . 8 (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2524anabss4 665 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2625ord 862 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵 → (𝐵𝐴) = 𝐵))
27 dfss2 3962 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2826, 27imbitrrdi 251 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵𝐵𝐴))
2919, 28orim12d 962 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵) → (𝐴𝐵𝐵𝐴)))
3011, 29mpd 15 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
31 sspsstri 4098 . . 3 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3230, 31sylib 217 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
33 ordelpss 6399 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
34 biidd 261 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
35 ordelpss 6399 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
3635ancoms 457 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
3733, 34, 363orbi123d 1431 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
3832, 37mpbird 256 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3o 1083   = wceq 1533  wcel 2098  cin 3943  wss 3944  wpss 3945  Ord word 6370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374
This theorem is referenced by:  ordtri1  6404  epweon  7778  epweonALT  7779  ordeleqon  7785  poseq  8163  soseq  8164  smo11  8385  smoord  8386  omopth2  8605  ttrcltr  9746  r111  9805  tcrank  9914  domtriomlem  10472  axdc3lem2  10481  zorn2lem6  10531  grur1  10850  nosepon  27649  addsproplem7  27943  negsproplem7  27997  mulsproplem13  28083  mulsproplem14  28084  oneltri  42830
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