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Theorem ordpwsucss 4227
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 4057 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if  On then both  U. suc (onunisuci 4119) and  U. {  On  |  C_  } (onuniss2 4187).

Constructively  ~P  i^i  On and  suc cannot be shown to be equivalent (as proved at ordpwsucexmid 4230). (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 4225 . . . . 5  Ord  Ord  suc
2 ordelon 4069 . . . . . 6  Ord  suc  suc  On
32ex 108 . . . . 5  Ord 
suc  suc  On
41, 3sylbi 114 . . . 4  Ord 
suc  On
5 ordtr 4064 . . . . 5  Ord  Tr
6 trsucss 4110 . . . . 5  Tr  suc 
C_
75, 6syl 14 . . . 4  Ord 
suc  C_
84, 7jcad 291 . . 3  Ord 
suc  On  C_
9 elin 3103 . . . 4  ~P  i^i  On  ~P  On
10 selpw 3341 . . . . 5  ~P  C_
1110anbi2ci 435 . . . 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 2928 1  Ord  suc  C_  ~P  i^i  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1374    i^i cin 2893    C_ wss 2894   ~Pcpw 3334   Tr wtr 3828   Ord word 4048   Oncon0 4049   suc csuc 4051
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by: (None)
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