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Theorem ordpwsuc 6567
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 3614 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
2 selpw 3947 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
32anbi2ci 694 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  On ) 
<->  ( x  e.  On  /\  x  C_  A )
)
41, 3bitri 249 . . 3  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  On  /\  x  C_  A ) )
5 ordsssuc 4891 . . . . . 6  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  C_  A  <->  x  e.  suc  A ) )
65expcom 433 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  On  ->  ( x  C_  A  <->  x  e.  suc  A ) ) )
76pm5.32d 637 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  ( x  e.  On  /\  x  e. 
suc  A ) ) )
8 simpr 459 . . . . 5  |-  ( ( x  e.  On  /\  x  e.  suc  A )  ->  x  e.  suc  A )
9 ordsuc 6566 . . . . . . 7  |-  ( Ord 
A  <->  Ord  suc  A )
10 ordelon 4829 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
1110ex 432 . . . . . . 7  |-  ( Ord 
suc  A  ->  ( x  e.  suc  A  ->  x  e.  On )
)
129, 11sylbi 195 . . . . . 6  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  x  e.  On ) )
1312ancrd 552 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  (
x  e.  On  /\  x  e.  suc  A ) ) )
148, 13impbid2 204 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  suc  A )  <-> 
x  e.  suc  A
) )
157, 14bitrd 253 . . 3  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  x  e.  suc  A ) )
164, 15syl5bb 257 . 2  |-  ( Ord 
A  ->  ( x  e.  ( ~P A  i^i  On )  <->  x  e.  suc  A ) )
1716eqrdv 2389 1  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836    i^i cin 3401    C_ wss 3402   ~Pcpw 3940   Ord word 4804   Oncon0 4805   suc csuc 4807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-tr 4474  df-eprel 4718  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-suc 4811
This theorem is referenced by:  onpwsuc  6568  orduniss2  6585
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