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Theorem pwwf 7660
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 7657 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
2 sspwb 4348 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
31, 2sylib 189 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
4 rankdmr1 7654 . . . . . . 7  |-  ( rank `  A )  e.  dom  R1
5 r1sucg 7622 . . . . . . 7  |-  ( (
rank `  A )  e.  dom  R1  ->  ( R1 `  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
) )
64, 5ax-mp 8 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
73, 6syl6sseqr 3332 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
8 fvex 5676 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
98elpw2 4299 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
107, 9sylibr 204 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
11 r1funlim 7619 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1211simpri 449 . . . . . . 7  |-  Lim  dom  R1
13 limsuc 4763 . . . . . . 7  |-  ( Lim 
dom  R1  ->  ( (
rank `  A )  e.  dom  R1  <->  suc  ( rank `  A )  e.  dom  R1 ) )
1412, 13ax-mp 8 . . . . . 6  |-  ( (
rank `  A )  e.  dom  R1  <->  suc  ( rank `  A )  e.  dom  R1 )
154, 14mpbi 200 . . . . 5  |-  suc  ( rank `  A )  e. 
dom  R1
16 r1sucg 7622 . . . . 5  |-  ( suc  ( rank `  A
)  e.  dom  R1  ->  ( R1 `  suc  suc  ( rank `  A
) )  =  ~P ( R1 `  suc  ( rank `  A ) ) )
1715, 16ax-mp 8 . . . 4  |-  ( R1
`  suc  suc  ( rank `  A ) )  =  ~P ( R1 `  suc  ( rank `  A
) )
1810, 17syl6eleqr 2472 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
19 r1elwf 7649 . . 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 7658 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
22 elex 2901 . . . . 5  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  e.  _V )
23 pwexb 4687 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 204 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  _V )
25 pwidg 3748 . . . 4  |-  ( A  e.  _V  ->  A  e.  ~P A )
2624, 25syl 16 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  ~P A
)
2721, 26sseldd 3286 . 2  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2820, 27impbii 181 1  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2893    C_ wss 3257   ~Pcpw 3736   U.cuni 3951   Oncon0 4516   Lim wlim 4517   suc csuc 4518   dom cdm 4812   "cima 4815   Fun wfun 5382   ` cfv 5388   R1cr1 7615   rankcrnk 7616
This theorem is referenced by:  snwf  7662  uniwf  7672  rankpwi  7676  r1pw  7698  r1pwcl  7700  dfac12r  7953  wfgru  8618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-recs 6563  df-rdg 6598  df-r1 7617  df-rank 7618
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