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Theorem r1pwcl 7535
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 7484 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
2 elfvdm 5570 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
31, 2jca 518 . . 3  |-  ( A  e.  ( R1 `  B )  ->  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )
43a1i 10 . 2  |-  ( Lim 
B  ->  ( A  e.  ( R1 `  B
)  ->  ( A  e.  U. ( R1 " On )  /\  B  e. 
dom  R1 ) ) )
5 r1elwf 7484 . . . . 5  |-  ( ~P A  e.  ( R1
`  B )  ->  ~P A  e.  U. ( R1 " On ) )
6 pwwf 7495 . . . . 5  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
75, 6sylibr 203 . . . 4  |-  ( ~P A  e.  ( R1
`  B )  ->  A  e.  U. ( R1 " On ) )
8 elfvdm 5570 . . . 4  |-  ( ~P A  e.  ( R1
`  B )  ->  B  e.  dom  R1 )
97, 8jca 518 . . 3  |-  ( ~P A  e.  ( R1
`  B )  -> 
( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )
109a1i 10 . 2  |-  ( Lim 
B  ->  ( ~P A  e.  ( R1 `  B )  ->  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) ) )
11 limsuc 4656 . . . . . 6  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
1211adantr 451 . . . . 5  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
13 rankpwi 7511 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
1413ad2antrl 708 . . . . . 6  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( rank `  ~P A )  =  suc  ( rank `  A
) )
1514eleq1d 2362 . . . . 5  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
1612, 15bitr4d 247 . . . 4  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
17 rankr1ag 7490 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
1817adantl 452 . . . 4  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
19 rankr1ag 7490 . . . . . 6  |-  ( ( ~P A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ~P A  e.  ( R1 `  B )  <->  ( rank `  ~P A )  e.  B ) )
206, 19sylanb 458 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ~P A  e.  ( R1 `  B
)  <->  ( rank `  ~P A )  e.  B
) )
2120adantl 452 . . . 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 276 . . 3  |-  ( ( Lim  B  /\  ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 ) )  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) )
2322ex 423 . 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 343 1  |-  ( Lim 
B  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ~Pcpw 3638   U.cuni 3843   Oncon0 4408   Lim wlim 4409   suc csuc 4410   dom cdm 4705   "cima 4708   ` cfv 5271   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  r1limwun  8374
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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