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Theorem ordpwsucss 4581
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 4386 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 4447) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4526).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4584). (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 4577 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
2 ordelon 4398 . . . . . 6  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
32ex 115 . . . . 5  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
41, 3sylbi 121 . . . 4  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
5 ordtr 4393 . . . . 5  |-  ( Ord 
A  ->  Tr  A
)
6 trsucss 4438 . . . . 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 307 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  C_  A ) ) )
9 elin 3333 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
10 velpw 3597 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
1110anbi2ci 459 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
129, 11bitri 184 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
138, 12imbitrrdi 162 . 2  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  ( ~P A  i^i  On ) ) )
1413ssrdv 3176 1  |-  ( Ord 
A  ->  suc  A  C_  ( ~P A  i^i  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160    i^i cin 3143    C_ wss 3144   ~Pcpw 3590   Tr wtr 4116   Ord word 4377   Oncon0 4378   suc csuc 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386
This theorem is referenced by: (None)
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