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Theorem ordpwsucss 4243
Description: The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of  ~P  i^i  On as another possible definition of successor, which would be equivalent to df-suc 4074 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if  On then both  U. suc (onunisuci 4135) and  U. {  On  |  C_  } (onuniss2 4203).

Constructively  ~P  i^i  On and  suc cannot be shown to be equivalent (as proved at ordpwsucexmid 4246). (Contributed by Jim Kingdon, 21-Jul-2019.)

Assertion
Ref Expression
ordpwsucss  Ord  suc  C_  ~P  i^i  On

Proof of Theorem ordpwsucss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ordsuc 4241 . . . . 5  Ord  Ord  suc
2 ordelon 4086 . . . . . 6  Ord  suc  suc  On
32ex 108 . . . . 5  Ord 
suc  suc  On
41, 3sylbi 114 . . . 4  Ord 
suc  On
5 ordtr 4081 . . . . 5  Ord  Tr
6 trsucss 4126 . . . . 5  Tr  suc 
C_
75, 6syl 14 . . . 4  Ord 
suc  C_
84, 7jcad 291 . . 3  Ord 
suc  On  C_
9 elin 3120 . . . 4  ~P  i^i  On  ~P  On
10 selpw 3358 . . . . 5  ~P  C_
1110anbi2ci 432 . . . 4  ~P  On  On  C_
129, 11bitri 173 . . 3  ~P  i^i  On  On  C_
138, 12syl6ibr 151 . 2  Ord 
suc  ~P  i^i  On
1413ssrdv 2945 1  Ord  suc  C_  ~P  i^i  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1390    i^i cin 2910    C_ wss 2911   ~Pcpw 3351   Tr wtr 3845   Ord word 4065   Oncon0 4066   suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
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