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Theorem onsucelsucexmid 4283
Description: The converse of onsucelsucr 4262 implies excluded middle. On the other hand, if  y is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4262 does hold, as seen at nnsucelsuc 6101. (Contributed by Jim Kingdon, 2-Aug-2019.)
Hypothesis
Ref Expression
onsucelsucexmid.1  |-  A. x  e.  On  A. y  e.  On  ( x  e.  y  ->  suc  x  e. 
suc  y )
Assertion
Ref Expression
onsucelsucexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x, y

Proof of Theorem onsucelsucexmid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 onsucelsucexmidlem1 4281 . . . 4  |-  (/)  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
2 0elon 4157 . . . . . 6  |-  (/)  e.  On
3 onsucelsucexmidlem 4282 . . . . . 6  |-  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  e.  On
42, 3pm3.2i 261 . . . . 5  |-  ( (/)  e.  On  /\  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  e.  On )
5 onsucelsucexmid.1 . . . . 5  |-  A. x  e.  On  A. y  e.  On  ( x  e.  y  ->  suc  x  e. 
suc  y )
6 eleq1 2116 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
7 suceq 4167 . . . . . . . 8  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
87eleq1d 2122 . . . . . . 7  |-  ( x  =  (/)  ->  ( suc  x  e.  suc  y  <->  suc  (/)  e.  suc  y ) )
96, 8imbi12d 227 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  y  ->  suc  x  e.  suc  y
)  <->  ( (/)  e.  y  ->  suc  (/)  e.  suc  y ) ) )
10 eleq2 2117 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( (/)  e.  y  <->  (/) 
e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } ) )
11 suceq 4167 . . . . . . . 8  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  y  =  suc  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )
1211eleq2d 2123 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  suc  y 
<->  suc  (/)  e.  suc  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1310, 12imbi12d 227 . . . . . 6  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( ( (/)  e.  y  ->  suc  (/)  e.  suc  y )  <->  ( (/)  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) } ) ) )
149, 13rspc2va 2686 . . . . 5  |-  ( ( ( (/)  e.  On  /\ 
{ z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  e.  On )  /\  A. x  e.  On  A. y  e.  On  (
x  e.  y  ->  suc  x  e.  suc  y
) )  ->  ( (/) 
e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
154, 5, 14mp2an 410 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
161, 15ax-mp 7 . . 3  |-  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
17 elsuci 4168 . . 3  |-  ( suc  (/)  e.  suc  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1816, 17ax-mp 7 . 2  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
19 suc0 4176 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 p0ex 3967 . . . . . . 7  |-  { (/) }  e.  _V
2120prid2 3505 . . . . . 6  |-  { (/) }  e.  { (/) ,  { (/)
} }
2219, 21eqeltri 2126 . . . . 5  |-  suc  (/)  e.  { (/)
,  { (/) } }
23 eqeq1 2062 . . . . . . 7  |-  ( z  =  suc  (/)  ->  (
z  =  (/)  <->  suc  (/)  =  (/) ) )
2423orbi1d 715 . . . . . 6  |-  ( z  =  suc  (/)  ->  (
( z  =  (/)  \/ 
ph )  <->  ( suc  (/)  =  (/)  \/  ph )
) )
2524elrab3 2722 . . . . 5  |-  ( suc  (/)  e.  { (/) ,  { (/)
} }  ->  ( suc  (/)  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( suc  (/)  =  (/)  \/ 
ph ) ) )
2622, 25ax-mp 7 . . . 4  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  ( suc  (/)  =  (/)  \/ 
ph ) )
27 0ex 3912 . . . . . . 7  |-  (/)  e.  _V
28 nsuceq0g 4183 . . . . . . 7  |-  ( (/)  e.  _V  ->  suc  (/)  =/=  (/) )
2927, 28ax-mp 7 . . . . . 6  |-  suc  (/)  =/=  (/)
30 df-ne 2221 . . . . . 6  |-  ( suc  (/)  =/=  (/)  <->  -.  suc  (/)  =  (/) )
3129, 30mpbi 137 . . . . 5  |-  -.  suc  (/)  =  (/)
32 pm2.53 651 . . . . 5  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ( -.  suc  (/)  =  (/)  ->  ph )
)
3331, 32mpi 15 . . . 4  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ph )
3426, 33sylbi 118 . . 3  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ph )
3519eqeq1i 2063 . . . . 5  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  { (/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
3619eqeq1i 2063 . . . . . . . 8  |-  ( suc  (/)  =  (/)  <->  { (/) }  =  (/) )
3731, 36mtbi 605 . . . . . . 7  |-  -.  { (/)
}  =  (/)
3820elsn 3419 . . . . . . 7  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
3937, 38mtbir 606 . . . . . 6  |-  -.  { (/)
}  e.  { (/) }
40 eleq2 2117 . . . . . 6  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  ( { (/) }  e.  { (/)
}  <->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
4139, 40mtbii 609 . . . . 5  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  -. 
{ (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4235, 41sylbi 118 . . . 4  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  { (/) }  e.  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
43 olc 642 . . . . 5  |-  ( ph  ->  ( { (/) }  =  (/) 
\/  ph ) )
44 eqeq1 2062 . . . . . . . 8  |-  ( z  =  { (/) }  ->  ( z  =  (/)  <->  { (/) }  =  (/) ) )
4544orbi1d 715 . . . . . . 7  |-  ( z  =  { (/) }  ->  ( ( z  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4645elrab3 2722 . . . . . 6  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4721, 46ax-mp 7 . . . . 5  |-  ( {
(/) }  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( { (/) }  =  (/)  \/  ph )
)
4843, 47sylibr 141 . . . 4  |-  ( ph  ->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4942, 48nsyl 568 . . 3  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  ph )
5034, 49orim12i 686 . 2  |-  ( ( suc  (/)  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )  ->  ( ph  \/  -.  ph )
)
5118, 50ax-mp 7 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409    =/= wne 2220   A.wral 2323   {crab 2327   _Vcvv 2574   (/)c0 3252   {csn 3403   {cpr 3404   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-ne 2221  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-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by:  ordsucunielexmid  4284
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