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Theorem ordtri3or 6210
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 6208 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordirr 6196 . . . . . 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 3935 . . . . . . 7 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
6 incom 4163 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
76eleq1i 2906 . . . . . . . 8 ((𝐴𝐵) ∈ 𝐵 ↔ (𝐵𝐴) ∈ 𝐵)
87anbi2i 625 . . . . . . 7 (((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
95, 8bitri 278 . . . . . 6 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐵𝐴) ∈ 𝐵))
104, 9xchnxbir 336 . . . . 5 (¬ (𝐴𝐵) ∈ (𝐴𝐵) ↔ (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
113, 10sylib 221 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 ∨ ¬ (𝐵𝐴) ∈ 𝐵))
12 inss1 4190 . . . . . . . . . 10 (𝐴𝐵) ⊆ 𝐴
13 ordsseleq 6207 . . . . . . . . . 10 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ⊆ 𝐴 ↔ ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴)))
1412, 13mpbii 236 . . . . . . . . 9 ((Ord (𝐴𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
151, 14sylan 583 . . . . . . . 8 (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1615anabss1 665 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) ∈ 𝐴 ∨ (𝐴𝐵) = 𝐴))
1716ord 861 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴 → (𝐴𝐵) = 𝐴))
18 df-ss 3936 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
1917, 18syl6ibr 255 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴𝐵) ∈ 𝐴𝐴𝐵))
20 ordin 6208 . . . . . . . . 9 ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵𝐴))
21 inss1 4190 . . . . . . . . . 10 (𝐵𝐴) ⊆ 𝐵
22 ordsseleq 6207 . . . . . . . . . 10 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ⊆ 𝐵 ↔ ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵)))
2321, 22mpbii 236 . . . . . . . . 9 ((Ord (𝐵𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2420, 23sylan 583 . . . . . . . 8 (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2524anabss4 666 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵𝐴) ∈ 𝐵 ∨ (𝐵𝐴) = 𝐵))
2625ord 861 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵𝐴) ∈ 𝐵 → (𝐵𝐴) = 𝐵))
27 df-ss 3936 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2826, 27syl6ibr 255 . . . . 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 4065 . . 3 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3230, 31sylib 221 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
33 ordelpss 6206 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
34 biidd 265 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
35 ordelpss 6206 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
3635ancoms 462 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
3733, 34, 363orbi123d 1432 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
3832, 37mpbird 260 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3o 1083   = wceq 1538  wcel 2115  cin 3918  wss 3919  wpss 3920  Ord word 6177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181
This theorem is referenced by:  ordtri1  6211  epweon  7491  ordeleqon  7497  smo11  7997  smoord  7998  omopth2  8206  r111  9201  tcrank  9310  domtriomlem  9862  axdc3lem2  9871  zorn2lem6  9921  grur1  10240  poseq  33155  soseq  33156  nosepon  33232
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