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Theorem onsucelsucexmid 4547
Description: The converse of onsucelsucr 4525 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 4525 does hold, as seen at nnsucelsuc 6516. (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 4545 . . . 4  |-  (/)  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
2 0elon 4410 . . . . . 6  |-  (/)  e.  On
3 onsucelsucexmidlem 4546 . . . . . 6  |-  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  e.  On
42, 3pm3.2i 272 . . . . 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 2252 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
7 suceq 4420 . . . . . . . 8  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
87eleq1d 2258 . . . . . . 7  |-  ( x  =  (/)  ->  ( suc  x  e.  suc  y  <->  suc  (/)  e.  suc  y ) )
96, 8imbi12d 234 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  y  ->  suc  x  e.  suc  y
)  <->  ( (/)  e.  y  ->  suc  (/)  e.  suc  y ) ) )
10 eleq2 2253 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( (/)  e.  y  <->  (/) 
e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } ) )
11 suceq 4420 . . . . . . . 8  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  y  =  suc  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )
1211eleq2d 2259 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  suc  y 
<->  suc  (/)  e.  suc  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1310, 12imbi12d 234 . . . . . 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 2870 . . . . 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 426 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
161, 15ax-mp 5 . . 3  |-  suc  (/)  e.  suc  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
17 elsuci 4421 . . 3  |-  ( suc  (/)  e.  suc  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1816, 17ax-mp 5 . 2  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
19 suc0 4429 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 p0ex 4206 . . . . . . 7  |-  { (/) }  e.  _V
2120prid2 3714 . . . . . 6  |-  { (/) }  e.  { (/) ,  { (/)
} }
2219, 21eqeltri 2262 . . . . 5  |-  suc  (/)  e.  { (/)
,  { (/) } }
23 eqeq1 2196 . . . . . . 7  |-  ( z  =  suc  (/)  ->  (
z  =  (/)  <->  suc  (/)  =  (/) ) )
2423orbi1d 792 . . . . . 6  |-  ( z  =  suc  (/)  ->  (
( z  =  (/)  \/ 
ph )  <->  ( suc  (/)  =  (/)  \/  ph )
) )
2524elrab3 2909 . . . . 5  |-  ( suc  (/)  e.  { (/) ,  { (/)
} }  ->  ( suc  (/)  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( suc  (/)  =  (/)  \/ 
ph ) ) )
2622, 25ax-mp 5 . . . 4  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  ( suc  (/)  =  (/)  \/ 
ph ) )
27 0ex 4145 . . . . . . 7  |-  (/)  e.  _V
28 nsuceq0g 4436 . . . . . . 7  |-  ( (/)  e.  _V  ->  suc  (/)  =/=  (/) )
2927, 28ax-mp 5 . . . . . 6  |-  suc  (/)  =/=  (/)
30 df-ne 2361 . . . . . 6  |-  ( suc  (/)  =/=  (/)  <->  -.  suc  (/)  =  (/) )
3129, 30mpbi 145 . . . . 5  |-  -.  suc  (/)  =  (/)
32 pm2.53 723 . . . . 5  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ( -.  suc  (/)  =  (/)  ->  ph )
)
3331, 32mpi 15 . . . 4  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ph )
3426, 33sylbi 121 . . 3  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ph )
3519eqeq1i 2197 . . . . 5  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  { (/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
3619eqeq1i 2197 . . . . . . . 8  |-  ( suc  (/)  =  (/)  <->  { (/) }  =  (/) )
3731, 36mtbi 671 . . . . . . 7  |-  -.  { (/)
}  =  (/)
3820elsn 3623 . . . . . . 7  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
3937, 38mtbir 672 . . . . . 6  |-  -.  { (/)
}  e.  { (/) }
40 eleq2 2253 . . . . . 6  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  ( { (/) }  e.  { (/)
}  <->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
4139, 40mtbii 675 . . . . 5  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  -. 
{ (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4235, 41sylbi 121 . . . 4  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  { (/) }  e.  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
43 olc 712 . . . . 5  |-  ( ph  ->  ( { (/) }  =  (/) 
\/  ph ) )
44 eqeq1 2196 . . . . . . . 8  |-  ( z  =  { (/) }  ->  ( z  =  (/)  <->  { (/) }  =  (/) ) )
4544orbi1d 792 . . . . . . 7  |-  ( z  =  { (/) }  ->  ( ( z  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4645elrab3 2909 . . . . . 6  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4721, 46ax-mp 5 . . . . 5  |-  ( {
(/) }  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( { (/) }  =  (/)  \/  ph )
)
4843, 47sylibr 134 . . . 4  |-  ( ph  ->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4942, 48nsyl 629 . . 3  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  ph )
5034, 49orim12i 760 . 2  |-  ( ( suc  (/)  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  \/  suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )  ->  ( ph  \/  -.  ph )
)
5118, 50ax-mp 5 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160    =/= wne 2360   A.wral 2468   {crab 2472   _Vcvv 2752   (/)c0 3437   {csn 3607   {cpr 3608   Oncon0 4381   suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389
This theorem is referenced by:  ordsucunielexmid  4548
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