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Theorem pwwf 8237
Description: A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
pwwf  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )

Proof of Theorem pwwf
StepHypRef Expression
1 r1rankidb 8234 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
2 sspwb 4702 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
31, 2sylib 196 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
4 rankdmr1 8231 . . . . . . 7  |-  ( rank `  A )  e.  dom  R1
5 r1sucg 8199 . . . . . . 7  |-  ( (
rank `  A )  e.  dom  R1  ->  ( R1 `  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
) )
64, 5ax-mp 5 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
73, 6syl6sseqr 3556 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
8 fvex 5882 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
98elpw2 4617 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
107, 9sylibr 212 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
11 r1funlim 8196 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1211simpri 462 . . . . . . 7  |-  Lim  dom  R1
13 limsuc 6679 . . . . . . 7  |-  ( Lim 
dom  R1  ->  ( (
rank `  A )  e.  dom  R1  <->  suc  ( rank `  A )  e.  dom  R1 ) )
1412, 13ax-mp 5 . . . . . 6  |-  ( (
rank `  A )  e.  dom  R1  <->  suc  ( rank `  A )  e.  dom  R1 )
154, 14mpbi 208 . . . . 5  |-  suc  ( rank `  A )  e. 
dom  R1
16 r1sucg 8199 . . . . 5  |-  ( suc  ( rank `  A
)  e.  dom  R1  ->  ( R1 `  suc  suc  ( rank `  A
) )  =  ~P ( R1 `  suc  ( rank `  A ) ) )
1715, 16ax-mp 5 . . . 4  |-  ( R1
`  suc  suc  ( rank `  A ) )  =  ~P ( R1 `  suc  ( rank `  A
) )
1810, 17syl6eleqr 2566 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
19 r1elwf 8226 . . 3  |-  ( ~P A  e.  ( R1
`  suc  suc  ( rank `  A ) )  ->  ~P A  e.  U. ( R1 " On ) )
2018, 19syl 16 . 2  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  U. ( R1 " On ) )
21 r1elssi 8235 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
22 elex 3127 . . . . 5  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  e.  _V )
23 pwexb 6606 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 212 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  _V )
25 pwidg 4029 . . . 4  |-  ( A  e.  _V  ->  A  e.  ~P A )
2624, 25syl 16 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  ~P A
)
2721, 26sseldd 3510 . 2  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2820, 27impbii 188 1  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   Oncon0 4884   Lim wlim 4885   suc csuc 4886   dom cdm 5005   "cima 5008   Fun wfun 5588   ` cfv 5594   R1cr1 8192   rankcrnk 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-rank 8195
This theorem is referenced by:  snwf  8239  uniwf  8249  rankpwi  8253  r1pw  8275  r1pwcl  8277  dfac12r  8538  wfgru  9206
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