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Theorem ordtriexmid 4294
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  |-  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )
Assertion
Ref Expression
ordtriexmid  |-  ( ph  \/  -.  ph )
Distinct variable groups:    x, y    ph, x
Allowed substitution hint:    ph( y)

Proof of Theorem ordtriexmid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 noel 3272 . . . 4  |-  -.  {
z  e.  { (/) }  |  ph }  e.  (/)
2 ordtriexmidlem 4292 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  On
3 eleq1 2145 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  e.  (/) 
<->  { z  e.  { (/)
}  |  ph }  e.  (/) ) )
4 eqeq1 2089 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  =  (/) 
<->  { z  e.  { (/)
}  |  ph }  =  (/) ) )
5 eleq2 2146 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
63, 4, 53orbi123d 1243 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x )  <-> 
( { z  e. 
{ (/) }  |  ph }  e.  (/)  \/  {
z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } ) ) )
7 0elon 4176 . . . . . . . 8  |-  (/)  e.  On
8 0ex 3926 . . . . . . . . 9  |-  (/)  e.  _V
9 eleq1 2145 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( y  e.  On  <->  (/)  e.  On ) )
109anbi2d 452 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( ( x  e.  On  /\  y  e.  On )  <->  ( x  e.  On  /\  (/) 
e.  On ) ) )
11 eleq2 2146 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( x  e.  y  <->  x  e.  (/) ) )
12 eqeq2 2092 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( x  =  y  <->  x  =  (/) ) )
13 eleq1 2145 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( y  e.  x  <->  (/)  e.  x
) )
1411, 12, 133orbi123d 1243 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) ) )
1510, 14imbi12d 232 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )  <->  ( (
x  e.  On  /\  (/) 
e.  On )  -> 
( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) ) ) )
16 ordtriexmid.1 . . . . . . . . . 10  |-  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )
1716rspec2 2455 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
188, 15, 17vtocl 2662 . . . . . . . 8  |-  ( ( x  e.  On  /\  (/) 
e.  On )  -> 
( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) )
197, 18mpan2 416 . . . . . . 7  |-  ( x  e.  On  ->  (
x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) )
206, 19vtoclga 2673 . . . . . 6  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  ( { z  e.  { (/) }  |  ph }  e.  (/)  \/  {
z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } ) )
212, 20ax-mp 7 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  (/) 
\/  { z  e. 
{ (/) }  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } )
22 3orass 923 . . . . 5  |-  ( ( { z  e.  { (/)
}  |  ph }  e.  (/)  \/  { z  e.  { (/) }  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } )  <-> 
( { z  e. 
{ (/) }  |  ph }  e.  (/)  \/  ( { z  e.  { (/)
}  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } ) ) )
2321, 22mpbi 143 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  e.  (/) 
\/  ( { z  e.  { (/) }  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
241, 23mtpor 1357 . . 3  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } )
25 ordtriexmidlem2 4293 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
268snid 3444 . . . . . 6  |-  (/)  e.  { (/)
}
27 biidd 170 . . . . . . 7  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
2827elrab3 2759 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
2926, 28ax-mp 7 . . . . 5  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
3029biimpi 118 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
3125, 30orim12i 709 . . 3  |-  ( ( { z  e.  { (/)
}  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } )  ->  ( -.  ph  \/  ph ) )
3224, 31ax-mp 7 . 2  |-  ( -. 
ph  \/  ph )
33 orcom 680 . 2  |-  ( (
ph  \/  -.  ph )  <->  ( -.  ph  \/  ph )
)
3432, 33mpbir 144 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    \/ w3o 919    = wceq 1285    e. wcel 1434   A.wral 2353   {crab 2357   (/)c0 3268   {csn 3417   Oncon0 4147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-nul 3925  ax-pow 3969
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-uni 3623  df-tr 3897  df-iord 4150  df-on 4152  df-suc 4155
This theorem is referenced by: (None)
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