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Theorem ordtriexmid 4210
Description: Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

Hypothesis
Ref Expression
ordtriexmid.1  On  On
Assertion
Ref Expression
ordtriexmid
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem ordtriexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 noel 3222 . . . 4  {  { (/) }  |  }  (/)
2 ordtriexmidlem 4208 . . . . . 6  {  { (/) }  |  }  On
3 eleq1 2097 . . . . . . . 8  { 
{ (/) }  |  }  (/)  {  { (/)
}  |  }  (/)
4 eqeq1 2043 . . . . . . . 8  { 
{ (/) }  |  }  (/)  {  { (/)
}  |  }  (/)
5 eleq2 2098 . . . . . . . 8  { 
{ (/) }  |  }  (/)  (/)  {  { (/)
}  |  }
63, 4, 53orbi123d 1205 . . . . . . 7  { 
{ (/) }  |  }  (/)  (/)  (/)  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
7 0elon 4095 . . . . . . . 8  (/)  On
8 0ex 3875 . . . . . . . . 9  (/)  _V
9 eleq1 2097 . . . . . . . . . . 11  (/)  On  (/)  On
109anbi2d 437 . . . . . . . . . 10  (/)  On  On  On  (/)  On
11 eleq2 2098 . . . . . . . . . . 11  (/)  (/)
12 eqeq2 2046 . . . . . . . . . . 11  (/)  (/)
13 eleq1 2097 . . . . . . . . . . 11  (/)  (/)
1411, 12, 133orbi123d 1205 . . . . . . . . . 10  (/)  (/)  (/)  (/)
1510, 14imbi12d 223 . . . . . . . . 9  (/)  On  On  On  (/)  On  (/)  (/)  (/)
16 ordtriexmid.1 . . . . . . . . . 10  On  On
1716rspec2 2402 . . . . . . . . 9  On  On
188, 15, 17vtocl 2602 . . . . . . . 8  On  (/)  On  (/)  (/)  (/)
197, 18mpan2 401 . . . . . . 7  On  (/)  (/)  (/)
206, 19vtoclga 2613 . . . . . 6  {  { (/) }  |  }  On  {  { (/)
}  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
212, 20ax-mp 7 . . . . 5  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
22 3orass 887 . . . . 5  {  { (/)
}  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
2321, 22mpbi 133 . . . 4  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
241, 23mtp-or 1314 . . 3  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
25 ordtriexmidlem2 4209 . . . 4  {  { (/) }  |  }  (/)
268snid 3394 . . . . . 6  (/)  { (/)
}
27 biidd 161 . . . . . . 7  (/)
2827elrab3 2693 . . . . . 6  (/)  { (/) }  (/)  {  { (/)
}  |  }
2926, 28ax-mp 7 . . . . 5  (/)  {  { (/)
}  |  }
3029biimpi 113 . . . 4  (/)  {  { (/)
}  |  }
3125, 30orim12i 675 . . 3  {  { (/)
}  |  }  (/)  (/)  {  { (/) }  |  }
3224, 31ax-mp 7 . 2
33 orcom 646 . 2
3432, 33mpbir 134 1
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   w3o 883   wceq 1242   wcel 1390  wral 2300   {crab 2304   (/)c0 3218   {csn 3367   Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
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