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Theorem r1pwcl 7763
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 7712 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
2 elfvdm 5749 . . . 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 7712 . . . . 5  |-  ( ~P A  e.  ( R1
`  B )  ->  ~P A  e.  U. ( R1 " On ) )
6 pwwf 7723 . . . . 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 5749 . . . 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 4821 . . . . . 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 7739 . . . . . . 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 2501 . . . . 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 7718 . . . . 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 7718 . . . . . 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 1652    e. wcel 1725   ~Pcpw 3791   U.cuni 4007   Oncon0 4573   Lim wlim 4574   suc csuc 4575   dom cdm 4870   "cima 4873   ` cfv 5446   R1cr1 7678   rankcrnk 7679
This theorem is referenced by:  r1limwun  8601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7680  df-rank 7681
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