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Theorem ordpwsuc 6631
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 3669 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
2 selpw 4000 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
32anbi2ci 696 . . . 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 4950 . . . . . 6  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  C_  A  <->  x  e.  suc  A ) )
65expcom 435 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  On  ->  ( x  C_  A  <->  x  e.  suc  A ) ) )
76pm5.32d 639 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  C_  A )  <->  ( x  e.  On  /\  x  e. 
suc  A ) ) )
8 simpr 461 . . . . 5  |-  ( ( x  e.  On  /\  x  e.  suc  A )  ->  x  e.  suc  A )
9 ordsuc 6630 . . . . . . 7  |-  ( Ord 
A  <->  Ord  suc  A )
10 ordelon 4888 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  e.  On )
1110ex 434 . . . . . . 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 554 . . . . 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 2438 1  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    i^i cin 3457    C_ wss 3458   ~Pcpw 3993   Ord word 4863   Oncon0 4864   suc csuc 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870
This theorem is referenced by:  onpwsuc  6632  orduniss2  6649
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