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Theorem ordpwsucexmid 4322
Description: The subset in ordpwsucss 4319 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 3945 . . . . 5  |-  (/)  e.  ~P { z  e.  { (/)
}  |  ph }
2 0elon 4157 . . . . 5  |-  (/)  e.  On
3 elin 3154 . . . . 5  |-  ( (/)  e.  ( ~P { z  e.  { (/) }  |  ph }  i^i  On )  <-> 
( (/)  e.  ~P {
z  e.  { (/) }  |  ph }  /\  (/) 
e.  On ) )
41, 2, 3mpbir2an 860 . . . 4  |-  (/)  e.  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
5 ordtriexmidlem 4273 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  On
6 suceq 4167 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { z  e.  { (/)
}  |  ph }
)
7 pweq 3390 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ~P x  =  ~P { z  e. 
{ (/) }  |  ph } )
87ineq1d 3165 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ~P x  i^i  On )  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On ) )
96, 8eqeq12d 2070 . . . . . 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 2645 . . . . 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 2129 . . 3  |-  (/)  e.  suc  { z  e.  { (/) }  |  ph }
14 elsuci 4168 . . 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 3912 . . . . . 6  |-  (/)  e.  _V
1716snid 3430 . . . . 5  |-  (/)  e.  { (/)
}
18 biidd 165 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1918elrab3 2722 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { z  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
2017, 19ax-mp 7 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
2120biimpi 117 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  ->  ph )
22 ordtriexmidlem2 4274 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
2322eqcoms 2059 . . 3  |-  ( (/)  =  { z  e.  { (/)
}  |  ph }  ->  -.  ph )
2421, 23orim12i 686 . 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 102    \/ wo 639    = wceq 1259    e. wcel 1409   A.wral 2323   {crab 2327    i^i cin 2944   (/)c0 3252   ~Pcpw 3387   {csn 3403   Oncon0 4128   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by: (None)
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