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Theorem pwwf 7735
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 7732 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
2 sspwb 4415 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
31, 2sylib 190 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
4 rankdmr1 7729 . . . . . . 7  |-  ( rank `  A )  e.  dom  R1
5 r1sucg 7697 . . . . . . 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 3397 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
8 fvex 5744 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
98elpw2 4366 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
107, 9sylibr 205 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
11 r1funlim 7694 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1211simpri 450 . . . . . . 7  |-  Lim  dom  R1
13 limsuc 4831 . . . . . . 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 201 . . . . 5  |-  suc  ( rank `  A )  e. 
dom  R1
16 r1sucg 7697 . . . . 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 2529 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
19 r1elwf 7724 . . 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 7733 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
22 elex 2966 . . . . 5  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  e.  _V )
23 pwexb 4755 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 205 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  _V )
25 pwidg 3813 . . . 4  |-  ( A  e.  _V  ->  A  e.  ~P A )
2624, 25syl 16 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  ~P A
)
2721, 26sseldd 3351 . 2  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2820, 27impbii 182 1  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   Oncon0 4583   Lim wlim 4584   suc csuc 4585   dom cdm 4880   "cima 4883   Fun wfun 5450   ` cfv 5456   R1cr1 7690   rankcrnk 7691
This theorem is referenced by:  snwf  7737  uniwf  7747  rankpwi  7751  r1pw  7773  r1pwcl  7775  dfac12r  8028  wfgru  8693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-r1 7692  df-rank 7693
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