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Theorem ordpwsuc 4606
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 3358 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
2 vex 2791 . . . . . 6  |-  x  e. 
_V
32elpw 3631 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
43anbi2ci 677 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
51, 4bitri 240 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
6 ordsssuc 4479 . . . . . 6  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  C_  A  <->  x  e.  suc  A ) )
76expcom 424 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  On  ->  ( x  C_  A  <->  x  e.  suc  A ) ) )
87pm5.32d 620 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  ( x  e.  On  /\  x  e. 
suc  A ) ) )
9 simpr 447 . . . . 5  |-  ( ( x  e.  On  /\  x  e.  suc  A )  ->  x  e.  suc  A )
10 ordsuc 4605 . . . . . . 7  |-  ( Ord 
A  <->  Ord  suc  A )
11 ordelon 4416 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
1211ex 423 . . . . . . 7  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
1310, 12sylbi 187 . . . . . 6  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
1413ancrd 537 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  e.  suc  A ) ) )
159, 14impbid2 195 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  suc  A )  <-> 
x  e.  suc  A
) )
168, 15bitrd 244 . . 3  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  x  e.  suc  A ) )
175, 16syl5bb 248 . 2  |-  ( Ord 
A  ->  ( x  e.  ( ~P A  i^i  On )  <->  x  e.  suc  A ) )
1817eqrdv 2281 1  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  onpwsuc  4607  orduniss2  4624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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