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Theorem ordpwsucexmid 4455
Description: The subset in ordpwsucss 4452 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 4058 . . . . 5  |-  (/)  e.  ~P { z  e.  { (/)
}  |  ph }
2 0elon 4284 . . . . 5  |-  (/)  e.  On
3 elin 3229 . . . . 5  |-  ( (/)  e.  ( ~P { z  e.  { (/) }  |  ph }  i^i  On )  <-> 
( (/)  e.  ~P {
z  e.  { (/) }  |  ph }  /\  (/) 
e.  On ) )
41, 2, 3mpbir2an 911 . . . 4  |-  (/)  e.  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
5 ordtriexmidlem 4405 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  On
6 suceq 4294 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { z  e.  { (/)
}  |  ph }
)
7 pweq 3483 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ~P x  =  ~P { z  e. 
{ (/) }  |  ph } )
87ineq1d 3246 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ~P x  i^i  On )  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On ) )
96, 8eqeq12d 2132 . . . . . 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 2735 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  suc  { z  e.  { (/) }  |  ph }  =  ( ~P { z  e.  { (/)
}  |  ph }  i^i  On ) )
125, 11ax-mp 5 . . . 4  |-  suc  {
z  e.  { (/) }  |  ph }  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
134, 12eleqtrri 2193 . . 3  |-  (/)  e.  suc  { z  e.  { (/) }  |  ph }
14 elsuci 4295 . . 3  |-  ( (/)  e.  suc  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  {
z  e.  { (/) }  |  ph }  \/  (/)  =  { z  e. 
{ (/) }  |  ph } ) )
1513, 14ax-mp 5 . 2  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  \/  (/)  =  { z  e.  { (/) }  |  ph } )
16 0ex 4025 . . . . . 6  |-  (/)  e.  _V
1716snid 3526 . . . . 5  |-  (/)  e.  { (/)
}
18 biidd 171 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1918elrab3 2814 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
2017, 19ax-mp 5 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
2120biimpi 119 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
22 ordtriexmidlem2 4406 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
2322eqcoms 2120 . . 3  |-  ( (/)  =  { z  e.  { (/)
}  |  ph }  ->  -.  ph )
2421, 23orim12i 733 . 2  |-  ( (
(/)  e.  { z  e.  { (/) }  |  ph }  \/  (/)  =  {
z  e.  { (/) }  |  ph } )  ->  ( ph  \/  -.  ph ) )
2515, 24ax-mp 5 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   A.wral 2393   {crab 2397    i^i cin 3040   (/)c0 3333   ~Pcpw 3480   {csn 3497   Oncon0 4255   suc csuc 4257
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by: (None)
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