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Theorem ordpwsucexmid 4554
Description: The subset in ordpwsucss 4551 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 4150 . . . . 5  |-  (/)  e.  ~P { z  e.  { (/)
}  |  ph }
2 0elon 4377 . . . . 5  |-  (/)  e.  On
3 elin 3310 . . . . 5  |-  ( (/)  e.  ( ~P { z  e.  { (/) }  |  ph }  i^i  On )  <-> 
( (/)  e.  ~P {
z  e.  { (/) }  |  ph }  /\  (/) 
e.  On ) )
41, 2, 3mpbir2an 937 . . . 4  |-  (/)  e.  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On )
5 ordtriexmidlem 4503 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  On
6 suceq 4387 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { z  e.  { (/)
}  |  ph }
)
7 pweq 3569 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ~P x  =  ~P { z  e. 
{ (/) }  |  ph } )
87ineq1d 3327 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ~P x  i^i  On )  =  ( ~P { z  e. 
{ (/) }  |  ph }  i^i  On ) )
96, 8eqeq12d 2185 . . . . . 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 2805 . . . . 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 2246 . . 3  |-  (/)  e.  suc  { z  e.  { (/) }  |  ph }
14 elsuci 4388 . . 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 4116 . . . . . 6  |-  (/)  e.  _V
1716snid 3614 . . . . 5  |-  (/)  e.  { (/)
}
18 biidd 171 . . . . . 6  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1918elrab3 2887 . . . . 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 4504 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
2322eqcoms 2173 . . 3  |-  ( (/)  =  { z  e.  { (/)
}  |  ph }  ->  -.  ph )
2421, 23orim12i 754 . 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 703    = wceq 1348    e. wcel 2141   A.wral 2448   {crab 2452    i^i cin 3120   (/)c0 3414   ~Pcpw 3566   {csn 3583   Oncon0 4348   suc csuc 4350
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by: (None)
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