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Theorem pwwf 8010
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 8007 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
2 sspwb 4538 . . . . . . 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 8004 . . . . . . 7  |-  ( rank `  A )  e.  dom  R1
5 r1sucg 7972 . . . . . . 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 3400 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
8 fvex 5698 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
98elpw2 4453 . . . . 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 7969 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1211simpri 459 . . . . . . 7  |-  Lim  dom  R1
13 limsuc 6459 . . . . . . 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 7972 . . . . 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 2532 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
19 r1elwf 7999 . . 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 8008 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
22 elex 2979 . . . . 5  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  e.  _V )
23 pwexb 6386 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 212 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  _V )
25 pwidg 3870 . . . 4  |-  ( A  e.  _V  ->  A  e.  ~P A )
2624, 25syl 16 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  ~P A
)
2721, 26sseldd 3354 . 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 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   Oncon0 4715   Lim wlim 4716   suc csuc 4717   dom cdm 4836   "cima 4839   Fun wfun 5409   ` cfv 5415   R1cr1 7965   rankcrnk 7966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967  df-rank 7968
This theorem is referenced by:  snwf  8012  uniwf  8022  rankpwi  8026  r1pw  8048  r1pwcl  8050  dfac12r  8311  wfgru  8979
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