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Theorem ordpwsucss 4319
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 4136 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 4197) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4266).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4322). (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 4315 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
2 ordelon 4148 . . . . . 6  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
32ex 112 . . . . 5  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
41, 3sylbi 118 . . . 4  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
5 ordtr 4143 . . . . 5  |-  ( Ord 
A  ->  Tr  A
)
6 trsucss 4188 . . . . 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 295 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  C_  A ) ) )
9 elin 3154 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
10 selpw 3394 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
1110anbi2ci 440 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
129, 11bitri 177 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
138, 12syl6ibr 155 . 2  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  ( ~P A  i^i  On ) ) )
1413ssrdv 2979 1  |-  ( Ord 
A  ->  suc  A  C_  ( ~P A  i^i  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    e. wcel 1409    i^i cin 2944    C_ wss 2945   ~Pcpw 3387   Tr wtr 3882   Ord word 4127   Oncon0 4128   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by: (None)
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