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Theorem r1pwcl 7707
Description: The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1pwcl  |-  ( Lim 
B  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) )

Proof of Theorem r1pwcl
StepHypRef Expression
1 r1elwf 7656 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
2 elfvdm 5698 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
31, 2jca 519 . . 3  |-  ( A  e.  ( R1 `  B )  ->  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )
43a1i 11 . 2  |-  ( Lim 
B  ->  ( A  e.  ( R1 `  B
)  ->  ( A  e.  U. ( R1 " On )  /\  B  e. 
dom  R1 ) ) )
5 r1elwf 7656 . . . . 5  |-  ( ~P A  e.  ( R1
`  B )  ->  ~P A  e.  U. ( R1 " On ) )
6 pwwf 7667 . . . . 5  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
75, 6sylibr 204 . . . 4  |-  ( ~P A  e.  ( R1
`  B )  ->  A  e.  U. ( R1 " On ) )
8 elfvdm 5698 . . . 4  |-  ( ~P A  e.  ( R1
`  B )  ->  B  e.  dom  R1 )
97, 8jca 519 . . 3  |-  ( ~P A  e.  ( R1
`  B )  -> 
( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )
109a1i 11 . 2  |-  ( Lim 
B  ->  ( ~P A  e.  ( R1 `  B )  ->  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) ) )
11 limsuc 4770 . . . . . 6  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
1211adantr 452 . . . . 5  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
13 rankpwi 7683 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
1413ad2antrl 709 . . . . . 6  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( rank `  ~P A )  =  suc  ( rank `  A
) )
1514eleq1d 2454 . . . . 5  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
1612, 15bitr4d 248 . . . 4  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
17 rankr1ag 7662 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
1817adantl 453 . . . 4  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
19 rankr1ag 7662 . . . . . 6  |-  ( ( ~P A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ~P A  e.  ( R1 `  B )  <->  ( rank `  ~P A )  e.  B ) )
206, 19sylanb 459 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ~P A  e.  ( R1 `  B
)  <->  ( rank `  ~P A )  e.  B
) )
2120adantl 453 . . . 4  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ~P A  e.  ( R1 `  B )  <->  ( rank `  ~P A )  e.  B ) )
2216, 18, 213bitr4d 277 . . 3  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) )
2322ex 424 . 2  |-  ( Lim 
B  ->  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) ) )
244, 10, 23pm5.21ndd 344 1  |-  ( Lim 
B  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   ~Pcpw 3743   U.cuni 3958   Oncon0 4523   Lim wlim 4524   suc csuc 4525   dom cdm 4819   "cima 4822   ` cfv 5395   R1cr1 7622   rankcrnk 7623
This theorem is referenced by:  r1limwun  8545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-r1 7624  df-rank 7625
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