MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtri3or Structured version   Visualization version   GIF version

Theorem ordtri3or 6416
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 6414 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordirr 6402 . . . . . 6 (Ord (𝐴𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
31, 2syl 17 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
4 ianor 984 . . . . . 6 (¬ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
5 elin 3967 . . . . . . 7 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
6 incom 4209 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
76eleq1i 2832 . . . . . . . 8 ((𝐴𝐵) ∈ 𝐵 ↔ (𝐵𝐴) ∈ 𝐵)
87anbi2i 623 . . . . . . 7 (((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
95, 8bitri 275 . . . . . 6 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
104, 9xchnxbir 333 . . . . 5 (¬ (𝐴𝐵) ∈ (𝐴𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
113, 10sylib 218 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
12 inss1 4237 . . . . . . . . . 10 (𝐴𝐵) ⊆ 𝐴
13 ordsseleq 6413 . . . . . . . . . 10 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ⊆ 𝐴 ↔ ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴)))
1412, 13mpbii 233 . . . . . . . . 9 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
151, 14sylan 580 . . . . . . . 8 (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1615anabss1 666 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1716ord 865 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 → (𝐴𝐵) = 𝐴))
18 dfss2 3969 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
1917, 18imbitrrdi 252 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴𝐴𝐵))
20 ordin 6414 . . . . . . . . 9 ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵𝐴))
21 inss1 4237 . . . . . . . . . 10 (𝐵𝐴) ⊆ 𝐵
22 ordsseleq 6413 . . . . . . . . . 10 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ⊆ 𝐵 ↔ ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵)))
2321, 22mpbii 233 . . . . . . . . 9 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2420, 23sylan 580 . . . . . . . 8 (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2524anabss4 667 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2625ord 865 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵 → (𝐵𝐴) = 𝐵))
27 dfss2 3969 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2826, 27imbitrrdi 252 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵𝐵𝐴))
2919, 28orim12d 967 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵) → (𝐴𝐵𝐵𝐴)))
3011, 29mpd 15 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
31 sspsstri 4105 . . 3 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3230, 31sylib 218 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
33 ordelpss 6412 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
34 biidd 262 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
35 ordelpss 6412 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
3635ancoms 458 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
3733, 34, 363orbi123d 1437 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
3832, 37mpbird 257 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3o 1086   = wceq 1540  wcel 2108  cin 3950  wss 3951  wpss 3952  Ord word 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by:  ordtri1  6417  epweon  7795  epweonALT  7796  ordeleqon  7802  poseq  8183  soseq  8184  smo11  8404  smoord  8405  omopth2  8622  ttrcltr  9756  r111  9815  tcrank  9924  domtriomlem  10482  axdc3lem2  10491  zorn2lem6  10541  grur1  10860  nosepon  27710  addsproplem7  28008  negsproplem7  28066  mulsproplem13  28154  mulsproplem14  28155  oneltri  43270
  Copyright terms: Public domain W3C validator