MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1pwcl Unicode version

Theorem r1pwcl 7519
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 7468 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
2 elfvdm 5554 . . . 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 7468 . . . . 5  |-  ( ~P A  e.  ( R1
`  B )  ->  ~P A  e.  U. ( R1 " On ) )
6 pwwf 7479 . . . . 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 5554 . . . 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 4640 . . . . . 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 7495 . . . . . . 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 2349 . . . . 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 7474 . . . . 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 7474 . . . . . 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 1623    e. wcel 1684   ~Pcpw 3625   U.cuni 3827   Oncon0 4392   Lim wlim 4393   suc csuc 4394   dom cdm 4689   "cima 4692   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  r1limwun  8358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
  Copyright terms: Public domain W3C validator