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Theorem onsucelsucexmid 4449
Description: The converse of onsucelsucr 4428 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 4428 does hold, as seen at nnsucelsuc 6391. (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 4447 . . . 4  |-  (/)  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }
2 0elon 4318 . . . . . 6  |-  (/)  e.  On
3 onsucelsucexmidlem 4448 . . . . . 6  |-  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  e.  On
42, 3pm3.2i 270 . . . . 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 2203 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
7 suceq 4328 . . . . . . . 8  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
87eleq1d 2209 . . . . . . 7  |-  ( x  =  (/)  ->  ( suc  x  e.  suc  y  <->  suc  (/)  e.  suc  y ) )
96, 8imbi12d 233 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  y  ->  suc  x  e.  suc  y
)  <->  ( (/)  e.  y  ->  suc  (/)  e.  suc  y ) ) )
10 eleq2 2204 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( (/)  e.  y  <->  (/) 
e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } ) )
11 suceq 4328 . . . . . . . 8  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  suc  y  =  suc  { z  e.  { (/)
,  { (/) } }  |  ( z  =  (/)  \/  ph ) } )
1211eleq2d 2210 . . . . . . 7  |-  ( y  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ( suc  (/)  e.  suc  y 
<->  suc  (/)  e.  suc  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
1310, 12imbi12d 233 . . . . . 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 2804 . . . . 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 423 . . . 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 4329 . . 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 4337 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 p0ex 4116 . . . . . . 7  |-  { (/) }  e.  _V
2120prid2 3634 . . . . . 6  |-  { (/) }  e.  { (/) ,  { (/)
} }
2219, 21eqeltri 2213 . . . . 5  |-  suc  (/)  e.  { (/)
,  { (/) } }
23 eqeq1 2147 . . . . . . 7  |-  ( z  =  suc  (/)  ->  (
z  =  (/)  <->  suc  (/)  =  (/) ) )
2423orbi1d 781 . . . . . 6  |-  ( z  =  suc  (/)  ->  (
( z  =  (/)  \/ 
ph )  <->  ( suc  (/)  =  (/)  \/  ph )
) )
2524elrab3 2842 . . . . 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 4059 . . . . . . 7  |-  (/)  e.  _V
28 nsuceq0g 4344 . . . . . . 7  |-  ( (/)  e.  _V  ->  suc  (/)  =/=  (/) )
2927, 28ax-mp 5 . . . . . 6  |-  suc  (/)  =/=  (/)
30 df-ne 2310 . . . . . 6  |-  ( suc  (/)  =/=  (/)  <->  -.  suc  (/)  =  (/) )
3129, 30mpbi 144 . . . . 5  |-  -.  suc  (/)  =  (/)
32 pm2.53 712 . . . . 5  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ( -.  suc  (/)  =  (/)  ->  ph )
)
3331, 32mpi 15 . . . 4  |-  ( ( suc  (/)  =  (/)  \/  ph )  ->  ph )
3426, 33sylbi 120 . . 3  |-  ( suc  (/)  e.  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  ph )
3519eqeq1i 2148 . . . . 5  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) } 
<->  { (/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
3619eqeq1i 2148 . . . . . . . 8  |-  ( suc  (/)  =  (/)  <->  { (/) }  =  (/) )
3731, 36mtbi 660 . . . . . . 7  |-  -.  { (/)
}  =  (/)
3820elsn 3544 . . . . . . 7  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
3937, 38mtbir 661 . . . . . 6  |-  -.  { (/)
}  e.  { (/) }
40 eleq2 2204 . . . . . 6  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  ( { (/) }  e.  { (/)
}  <->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } ) )
4139, 40mtbii 664 . . . . 5  |-  ( {
(/) }  =  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  ->  -. 
{ (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4235, 41sylbi 120 . . . 4  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  { (/) }  e.  { z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
43 olc 701 . . . . 5  |-  ( ph  ->  ( { (/) }  =  (/) 
\/  ph ) )
44 eqeq1 2147 . . . . . . . 8  |-  ( z  =  { (/) }  ->  ( z  =  (/)  <->  { (/) }  =  (/) ) )
4544orbi1d 781 . . . . . . 7  |-  ( z  =  { (/) }  ->  ( ( z  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4645elrab3 2842 . . . . . 6  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
4721, 46ax-mp 5 . . . . 5  |-  ( {
(/) }  e.  { z  e.  { (/) ,  { (/)
} }  |  ( z  =  (/)  \/  ph ) }  <->  ( { (/) }  =  (/)  \/  ph )
)
4843, 47sylibr 133 . . . 4  |-  ( ph  ->  { (/) }  e.  {
z  e.  { (/) ,  { (/) } }  | 
( z  =  (/)  \/ 
ph ) } )
4942, 48nsyl 618 . . 3  |-  ( suc  (/)  =  { z  e. 
{ (/) ,  { (/) } }  |  ( z  =  (/)  \/  ph ) }  ->  -.  ph )
5034, 49orim12i 749 . 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 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481    =/= wne 2309   A.wral 2417   {crab 2421   _Vcvv 2687   (/)c0 3364   {csn 3528   {cpr 3529   Oncon0 4289   suc csuc 4291
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-uni 3741  df-tr 4031  df-iord 4292  df-on 4294  df-suc 4297
This theorem is referenced by:  ordsucunielexmid  4450
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