ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordtriexmid Structured version   Unicode version

Theorem ordtriexmid 4190
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 3201 . . . 4  {  { (/) }  |  }  (/)
2 ordtriexmidlem 4188 . . . . . 6  {  { (/) }  |  }  On
3 eleq1 2078 . . . . . . . 8  { 
{ (/) }  |  }  (/)  {  { (/)
}  |  }  (/)
4 eqeq1 2024 . . . . . . . 8  { 
{ (/) }  |  }  (/)  {  { (/)
}  |  }  (/)
5 eleq2 2079 . . . . . . . 8  { 
{ (/) }  |  }  (/)  (/)  {  { (/)
}  |  }
63, 4, 53orbi123d 1189 . . . . . . 7  { 
{ (/) }  |  }  (/)  (/)  (/)  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
7 0elon 4074 . . . . . . . 8  (/)  On
8 0ex 3854 . . . . . . . . 9  (/)  _V
9 eleq1 2078 . . . . . . . . . . 11  (/)  On  (/)  On
109anbi2d 440 . . . . . . . . . 10  (/)  On  On  On  (/)  On
11 eleq2 2079 . . . . . . . . . . 11  (/)  (/)
12 eqeq2 2027 . . . . . . . . . . 11  (/)  (/)
13 eleq1 2078 . . . . . . . . . . 11  (/)  (/)
1411, 12, 133orbi123d 1189 . . . . . . . . . 10  (/)  (/)  (/)  (/)
1510, 14imbi12d 223 . . . . . . . . 9  (/)  On  On  On  (/)  On  (/)  (/)  (/)
16 ordtriexmid.1 . . . . . . . . . 10  On  On
1716rspec2 2382 . . . . . . . . 9  On  On
188, 15, 17vtocl 2581 . . . . . . . 8  On  (/)  On  (/)  (/)  (/)
197, 18mpan2 403 . . . . . . 7  On  (/)  (/)  (/)
206, 19vtoclga 2592 . . . . . 6  {  { (/) }  |  }  On  {  { (/)
}  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
212, 20ax-mp 7 . . . . 5  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
22 3orass 874 . . . . 5  {  { (/)
}  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
2321, 22mpbi 133 . . . 4  {  { (/) }  |  }  (/)  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
241, 23mtp-or 1298 . . 3  {  { (/) }  |  }  (/)  (/)  {  { (/) }  |  }
25 ordtriexmidlem2 4189 . . . 4  {  { (/) }  |  }  (/)
268snid 3373 . . . . . 6  (/)  { (/)
}
27 biidd 161 . . . . . . 7  (/)
2827elrab3 2672 . . . . . 6  (/)  { (/) }  (/)  {  { (/)
}  |  }
2926, 28ax-mp 7 . . . . 5  (/)  {  { (/)
}  |  }
3029biimpi 113 . . . 4  (/)  {  { (/)
}  |  }
3125, 30orim12i 663 . . 3  {  { (/)
}  |  }  (/)  (/)  {  { (/) }  |  }
3224, 31ax-mp 7 . 2
33 orcom 634 . 2
3432, 33mpbir 134 1
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 616   w3o 870   wceq 1226   wcel 1370  wral 2280   {crab 2284   (/)c0 3197   {csn 3346   Oncon0 4045
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897
This theorem depends on definitions:  df-bi 110  df-3or 872  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator