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Theorem pwwf 7697
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 7694 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
2 sspwb 4381 . . . . . . 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 7691 . . . . . . 7  |-  ( rank `  A )  e.  dom  R1
5 r1sucg 7659 . . . . . . 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 3363 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
8 fvex 5709 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
98elpw2 4332 . . . . 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 7656 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1211simpri 449 . . . . . . 7  |-  Lim  dom  R1
13 limsuc 4796 . . . . . . 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 7659 . . . . 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 2503 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
19 r1elwf 7686 . . 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 7695 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
22 elex 2932 . . . . 5  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  e.  _V )
23 pwexb 4720 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 204 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  _V )
25 pwidg 3779 . . . 4  |-  ( A  e.  _V  ->  A  e.  ~P A )
2624, 25syl 16 . . 3  |-  ( ~P A  e.  U. ( R1 " On )  ->  A  e.  ~P A
)
2721, 26sseldd 3317 . 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 1721   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767   U.cuni 3983   Oncon0 4549   Lim wlim 4550   suc csuc 4551   dom cdm 4845   "cima 4848   Fun wfun 5415   ` cfv 5421   R1cr1 7652   rankcrnk 7653
This theorem is referenced by:  snwf  7699  uniwf  7709  rankpwi  7713  r1pw  7735  r1pwcl  7737  dfac12r  7990  wfgru  8655
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-r1 7654  df-rank 7655
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