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Theorem ordpwsuc 4622
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 3371 . . . 4  |-  ( x  e.  ( ~P A  i^i  On )  <->  ( x  e.  ~P A  /\  x  e.  On ) )
2 vex 2804 . . . . . 6  |-  x  e. 
_V
32elpw 3644 . . . . 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 4495 . . . . . 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 4621 . . . . . . 7  |-  ( Ord 
A  <->  Ord  suc  A )
11 ordelon 4432 . . . . . . . 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 2294 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  onpwsuc  4623  orduniss2  4640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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