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Theorem ordpwsuc 4797
Description: The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)
Assertion
Ref Expression
ordpwsuc  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )

Proof of Theorem ordpwsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3532 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
2 vex 2961 . . . . . 6  |-  x  e. 
_V
32elpw 3807 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
43anbi2ci 679 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
51, 4bitri 242 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
6 ordsssuc 4670 . . . . . 6  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  C_  A  <->  x  e.  suc  A ) )
76expcom 426 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  On  ->  ( x  C_  A  <->  x  e.  suc  A ) ) )
87pm5.32d 622 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  ( x  e.  On  /\  x  e. 
suc  A ) ) )
9 simpr 449 . . . . 5  |-  ( ( x  e.  On  /\  x  e.  suc  A )  ->  x  e.  suc  A )
10 ordsuc 4796 . . . . . . 7  |-  ( Ord 
A  <->  Ord  suc  A )
11 ordelon 4607 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
1211ex 425 . . . . . . 7  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
1310, 12sylbi 189 . . . . . 6  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
1413ancrd 539 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  e.  suc  A ) ) )
159, 14impbid2 197 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  suc  A )  <-> 
x  e.  suc  A
) )
168, 15bitrd 246 . . 3  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  x  e.  suc  A ) )
175, 16syl5bb 250 . 2  |-  ( Ord 
A  ->  ( x  e.  ( ~P A  i^i  On )  <->  x  e.  suc  A ) )
1817eqrdv 2436 1  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   Ord word 4582   Oncon0 4583   suc csuc 4585
This theorem is referenced by:  onpwsuc  4798  orduniss2  4815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589
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