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Theorem ordpwsucss 4396
Description: The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of  ( ~P A  i^i  On ) as another possible definition of successor, which would be equivalent to df-suc 4207 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if  A  e.  On then both  U. suc  A  =  A (onunisuci 4268) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4342).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4399). (Contributed by Jim Kingdon, 21-Jul-2019.)

Assertion
Ref Expression
ordpwsucss  |-  ( Ord 
A  ->  suc  A  C_  ( ~P A  i^i  On ) )

Proof of Theorem ordpwsucss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordsuc 4392 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
2 ordelon 4219 . . . . . 6  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
32ex 114 . . . . 5  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
41, 3sylbi 120 . . . 4  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
5 ordtr 4214 . . . . 5  |-  ( Ord 
A  ->  Tr  A
)
6 trsucss 4259 . . . . 5  |-  ( Tr  A  ->  ( x  e.  suc  A  ->  x  C_  A ) )
75, 6syl 14 . . . 4  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  C_  A ) )
84, 7jcad 302 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  C_  A ) ) )
9 elin 3184 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
10 selpw 3440 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
1110anbi2ci 448 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
129, 11bitri 183 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
138, 12syl6ibr 161 . 2  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  ( ~P A  i^i  On ) ) )
1413ssrdv 3032 1  |-  ( Ord 
A  ->  suc  A  C_  ( ~P A  i^i  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1439    i^i cin 2999    C_ wss 3000   ~Pcpw 3433   Tr wtr 3942   Ord word 4198   Oncon0 4199   suc csuc 4201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207
This theorem is referenced by: (None)
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