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Theorem ordtriexmid 4432
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 3362 . . . 4  |-  -.  {
z  e.  { (/) }  |  ph }  e.  (/)
2 ordtriexmidlem 4430 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  e.  On
3 eleq1 2200 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  e.  (/) 
<->  { z  e.  { (/)
}  |  ph }  e.  (/) ) )
4 eqeq1 2144 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  =  (/) 
<->  { z  e.  { (/)
}  |  ph }  =  (/) ) )
5 eleq2 2201 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
63, 4, 53orbi123d 1289 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x )  <-> 
( { z  e. 
{ (/) }  |  ph }  e.  (/)  \/  {
z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } ) ) )
7 0elon 4309 . . . . . . . 8  |-  (/)  e.  On
8 0ex 4050 . . . . . . . . 9  |-  (/)  e.  _V
9 eleq1 2200 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( y  e.  On  <->  (/)  e.  On ) )
109anbi2d 459 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( ( x  e.  On  /\  y  e.  On )  <->  ( x  e.  On  /\  (/) 
e.  On ) ) )
11 eleq2 2201 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( x  e.  y  <->  x  e.  (/) ) )
12 eqeq2 2147 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( x  =  y  <->  x  =  (/) ) )
13 eleq1 2200 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( y  e.  x  <->  (/)  e.  x
) )
1411, 12, 133orbi123d 1289 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) ) )
1510, 14imbi12d 233 . . . . . . . . 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 2519 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
188, 15, 17vtocl 2735 . . . . . . . 8  |-  ( ( x  e.  On  /\  (/) 
e.  On )  -> 
( x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) )
197, 18mpan2 421 . . . . . . 7  |-  ( x  e.  On  ->  (
x  e.  (/)  \/  x  =  (/)  \/  (/)  e.  x
) )
206, 19vtoclga 2747 . . . . . 6  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  ( { z  e.  { (/) }  |  ph }  e.  (/)  \/  {
z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } ) )
212, 20ax-mp 5 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  (/) 
\/  { z  e. 
{ (/) }  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } )
22 3orass 965 . . . . 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 144 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  e.  (/) 
\/  ( { z  e.  { (/) }  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
241, 23mtpor 1403 . . 3  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
\/  (/)  e.  { z  e.  { (/) }  |  ph } )
25 ordtriexmidlem2 4431 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
268snid 3551 . . . . . 6  |-  (/)  e.  { (/)
}
27 biidd 171 . . . . . . 7  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
2827elrab3 2836 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
2926, 28ax-mp 5 . . . . 5  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
3029biimpi 119 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
3125, 30orim12i 748 . . 3  |-  ( ( { z  e.  { (/)
}  |  ph }  =  (/)  \/  (/)  e.  {
z  e.  { (/) }  |  ph } )  ->  ( -.  ph  \/  ph ) )
3224, 31ax-mp 5 . 2  |-  ( -. 
ph  \/  ph )
33 orcom 717 . 2  |-  ( (
ph  \/  -.  ph )  <->  ( -.  ph  \/  ph )
)
3432, 33mpbir 145 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   A.wral 2414   {crab 2418   (/)c0 3358   {csn 3522   Oncon0 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288
This theorem is referenced by: (None)
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