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Theorem onsucelsucexmid 4336
Description: The converse of onsucelsucr 4315 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 4315 does hold, as seen at nnsucelsuc 6234. (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 4334 . . . 4  |-  (/)  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
2 0elon 4210 . . . . . 6  |-  (/)  e.  On
3 onsucelsucexmidlem 4335 . . . . . 6  |-  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  e.  On
42, 3pm3.2i 266 . . . . 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 2150 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
7 suceq 4220 . . . . . . . 8  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
87eleq1d 2156 . . . . . . 7  |-  ( x  =  (/)  ->  ( suc  x  e.  suc  y  <->  suc  (/)  e.  suc  y ) )
96, 8imbi12d 232 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  y  ->  suc  x  e.  suc  y
)  <->  ( (/)  e.  y  ->  suc  (/)  e.  suc  y ) ) )
10 eleq2 2151 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( (/)  e.  y  <->  (/) 
e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } ) )
11 suceq 4220 . . . . . . . 8  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  y  =  suc  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )
1211eleq2d 2157 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  suc  y 
<->  suc  (/)  e.  suc  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1310, 12imbi12d 232 . . . . . 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 2734 . . . . 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 417 . . . 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 4221 . . 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 4229 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 p0ex 4014 . . . . . . 7  |-  { (/) }  e.  _V
2120prid2 3544 . . . . . 6  |-  { (/) }  e.  { (/) ,  { (/)
} }
2219, 21eqeltri 2160 . . . . 5  |-  suc  (/)  e.  { (/)
,  { (/) } }
23 eqeq1 2094 . . . . . . 7  |-  ( z  =  suc  (/)  ->  (
z  =  (/)  <->  suc  (/)  =  (/) ) )
2423orbi1d 740 . . . . . 6  |-  ( z  =  suc  (/)  ->  (
( z  =  (/)  \/ 
ph )  <->  ( suc  (/)  =  (/)  \/  ph )
) )
2524elrab3 2770 . . . . 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 3958 . . . . . . 7  |-  (/)  e.  _V
28 nsuceq0g 4236 . . . . . . 7  |-  ( (/)  e.  _V  ->  suc  (/)  =/=  (/) )
2927, 28ax-mp 7 . . . . . 6  |-  suc  (/)  =/=  (/)
30 df-ne 2256 . . . . . 6  |-  ( suc  (/)  =/=  (/)  <->  -.  suc  (/)  =  (/) )
3129, 30mpbi 143 . . . . 5  |-  -.  suc  (/)  =  (/)
32 pm2.53 676 . . . . 5  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ( -.  suc  (/)  =  (/)  ->  ph )
)
3331, 32mpi 15 . . . 4  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ph )
3426, 33sylbi 119 . . 3  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ph )
3519eqeq1i 2095 . . . . 5  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  { (/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
3619eqeq1i 2095 . . . . . . . 8  |-  ( suc  (/)  =  (/)  <->  { (/) }  =  (/) )
3731, 36mtbi 630 . . . . . . 7  |-  -.  { (/)
}  =  (/)
3820elsn 3457 . . . . . . 7  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
3937, 38mtbir 631 . . . . . 6  |-  -.  { (/)
}  e.  { (/) }
40 eleq2 2151 . . . . . 6  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  ( { (/) }  e.  { (/)
}  <->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
4139, 40mtbii 634 . . . . 5  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  -. 
{ (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4235, 41sylbi 119 . . . 4  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  { (/) }  e.  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
43 olc 667 . . . . 5  |-  ( ph  ->  ( { (/) }  =  (/) 
\/  ph ) )
44 eqeq1 2094 . . . . . . . 8  |-  ( z  =  { (/) }  ->  ( z  =  (/)  <->  { (/) }  =  (/) ) )
4544orbi1d 740 . . . . . . 7  |-  ( z  =  { (/) }  ->  ( ( z  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4645elrab3 2770 . . . . . 6  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4721, 46ax-mp 7 . . . . 5  |-  ( {
(/) }  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( { (/) }  =  (/)  \/  ph )
)
4843, 47sylibr 132 . . . 4  |-  ( ph  ->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4942, 48nsyl 593 . . 3  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  ph )
5034, 49orim12i 711 . 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 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438    =/= wne 2255   A.wral 2359   {crab 2363   _Vcvv 2619   (/)c0 3284   {csn 3441   {cpr 3442   Oncon0 4181   suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by:  ordsucunielexmid  4337
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