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Theorem ordpwsucexmid 4386
Description: The subset in ordpwsucss 4383 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
Hypothesis
Ref Expression
ordpwsucexmid.1  |-  A. x  e.  On  suc  x  =  ( ~P x  i^i 
On )
Assertion
Ref Expression
ordpwsucexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x

Proof of Theorem ordpwsucexmid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 0elpw 3999 . . . . 5  |-  (/)  e.  ~P { z  e.  { (/)
}  |  ph }
2 0elon 4219 . . . . 5  |-  (/)  e.  On
3 elin 3183 . . . . 5  |-  ( (/)  e.  ( ~P { z  e.  { (/) }  |  ph }  i^i  On )  <-> 
( (/)  e.  ~P {
z  e.  { (/) }  |  ph }  /\  (/) 
e.  On ) )
41, 2, 3mpbir2an 888 . . . 4  |-  (/)  e.  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
5 ordtriexmidlem 4336 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  On
6 suceq 4229 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { z  e.  { (/)
}  |  ph }
)
7 pweq 3432 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ~P x  =  ~P { z  e. 
{ (/) }  |  ph } )
87ineq1d 3200 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ~P x  i^i  On )  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On ) )
96, 8eqeq12d 2102 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( suc  x  =  ( ~P x  i^i  On )  <->  suc  { z  e.  { (/) }  |  ph }  =  ( ~P { z  e.  { (/)
}  |  ph }  i^i  On ) ) )
10 ordpwsucexmid.1 . . . . . 6  |-  A. x  e.  On  suc  x  =  ( ~P x  i^i 
On )
119, 10vtoclri 2694 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  suc  { z  e.  { (/) }  |  ph }  =  ( ~P { z  e.  { (/)
}  |  ph }  i^i  On ) )
125, 11ax-mp 7 . . . 4  |-  suc  {
z  e.  { (/) }  |  ph }  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
134, 12eleqtrri 2163 . . 3  |-  (/)  e.  suc  { z  e.  { (/) }  |  ph }
14 elsuci 4230 . . 3  |-  ( (/)  e.  suc  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  {
z  e.  { (/) }  |  ph }  \/  (/)  =  { z  e. 
{ (/) }  |  ph } ) )
1513, 14ax-mp 7 . 2  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  \/  (/)  =  { z  e.  { (/) }  |  ph } )
16 0ex 3966 . . . . . 6  |-  (/)  e.  _V
1716snid 3475 . . . . 5  |-  (/)  e.  { (/)
}
18 biidd 170 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1918elrab3 2772 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
2017, 19ax-mp 7 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
2120biimpi 118 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
22 ordtriexmidlem2 4337 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
2322eqcoms 2091 . . 3  |-  ( (/)  =  { z  e.  { (/)
}  |  ph }  ->  -.  ph )
2421, 23orim12i 711 . 2  |-  ( (
(/)  e.  { z  e.  { (/) }  |  ph }  \/  (/)  =  {
z  e.  { (/) }  |  ph } )  ->  ( ph  \/  -.  ph ) )
2515, 24ax-mp 7 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   A.wral 2359   {crab 2363    i^i cin 2998   (/)c0 3286   ~Pcpw 3429   {csn 3446   Oncon0 4190   suc csuc 4192
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198
This theorem is referenced by: (None)
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