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

Theorem ordtri3or 5793
Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (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 5791 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordirr 5779 . . . . . 6 (Ord (𝐴𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
31, 2syl 17 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
4 ianor 508 . . . . . 6 (¬ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
5 elin 3829 . . . . . . 7 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
6 incom 3838 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
76eleq1i 2721 . . . . . . . 8 ((𝐴𝐵) ∈ 𝐵 ↔ (𝐵𝐴) ∈ 𝐵)
87anbi2i 730 . . . . . . 7 (((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
95, 8bitri 264 . . . . . 6 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
104, 9xchnxbir 322 . . . . 5 (¬ (𝐴𝐵) ∈ (𝐴𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
113, 10sylib 208 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
12 inss1 3866 . . . . . . . . . 10 (𝐴𝐵) ⊆ 𝐴
13 ordsseleq 5790 . . . . . . . . . 10 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ⊆ 𝐴 ↔ ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴)))
1412, 13mpbii 223 . . . . . . . . 9 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
151, 14sylan 487 . . . . . . . 8 (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1615anabss1 872 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1716ord 391 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 → (𝐴𝐵) = 𝐴))
18 df-ss 3621 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
1917, 18syl6ibr 242 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴𝐴𝐵))
20 ordin 5791 . . . . . . . . 9 ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵𝐴))
21 inss1 3866 . . . . . . . . . 10 (𝐵𝐴) ⊆ 𝐵
22 ordsseleq 5790 . . . . . . . . . 10 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ⊆ 𝐵 ↔ ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵)))
2321, 22mpbii 223 . . . . . . . . 9 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2420, 23sylan 487 . . . . . . . 8 (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2524anabss4 873 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2625ord 391 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵 → (𝐵𝐴) = 𝐵))
27 df-ss 3621 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2826, 27syl6ibr 242 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵𝐵𝐴))
2919, 28orim12d 901 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → ((¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵) → (𝐴𝐵𝐵𝐴)))
3011, 29mpd 15 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
31 sspsstri 3742 . . 3 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3230, 31sylib 208 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
33 ordelpss 5789 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
34 biidd 252 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
35 ordelpss 5789 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
3635ancoms 468 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
3733, 34, 363orbi123d 1438 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
3832, 37mpbird 247 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1053   = wceq 1523  wcel 2030  cin 3606  wss 3607  wpss 3608  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  ordtri1  5794  ordtri3OLD  5798  ordon  7024  ordeleqon  7030  smo11  7506  smoord  7507  omopth2  7709  r111  8676  tcrank  8785  domtriomlem  9302  axdc3lem2  9311  zorn2lem6  9361  grur1  9680  poseq  31878  soseq  31879  nosepon  31943
  Copyright terms: Public domain W3C validator