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Theorem r1pw 8266
Description: A stronger property of  R1 than rankpw 8264. The latter merely proves that  R1 of the successor is a power set, but here we prove that if  A is in the cumulative hierarchy, then  ~P A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1pw  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )

Proof of Theorem r1pw
StepHypRef Expression
1 rankpwi 8244 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
21eleq1d 2512 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  ~P A )  e.  suc  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
3 eloni 4878 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
4 ordsucelsuc 6642 . . . . . . 7  |-  ( Ord 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
53, 4syl 16 . . . . . 6  |-  ( B  e.  On  ->  (
( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
65bicomd 201 . . . . 5  |-  ( B  e.  On  ->  ( suc  ( rank `  A
)  e.  suc  B  <->  (
rank `  A )  e.  B ) )
72, 6sylan9bb 699 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ( rank `  ~P A )  e.  suc  B  <-> 
( rank `  A )  e.  B ) )
8 pwwf 8228 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
98biimpi 194 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  U. ( R1 " On ) )
10 suceloni 6633 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
11 r1fnon 8188 . . . . . . 7  |-  R1  Fn  On
12 fndm 5670 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 5 . . . . . 6  |-  dom  R1  =  On
1410, 13syl6eleqr 2542 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  dom  R1 )
15 rankr1ag 8223 . . . . 5  |-  ( ( ~P A  e.  U. ( R1 " On )  /\  suc  B  e. 
dom  R1 )  ->  ( ~P A  e.  ( R1 `  suc  B )  <-> 
( rank `  ~P A )  e.  suc  B ) )
169, 14, 15syl2an 477 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ~P A  e.  ( R1 `  suc  B )  <->  ( rank `  ~P A )  e.  suc  B ) )
1713eleq2i 2521 . . . . 5  |-  ( B  e.  dom  R1  <->  B  e.  On )
18 rankr1ag 8223 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
1917, 18sylan2br 476 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <-> 
( rank `  A )  e.  B ) )
207, 16, 193bitr4rd 286 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
2120ex 434 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
22 r1elwf 8217 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
23 r1elwf 8217 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  U. ( R1 " On ) )
24 r1elssi 8226 . . . . . 6  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
2523, 24syl 16 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  C_  U. ( R1 " On ) )
26 ssid 3508 . . . . . 6  |-  A  C_  A
27 elex 3104 . . . . . . . 8  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
28 pwexb 6596 . . . . . . . 8  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2927, 28sylibr 212 . . . . . . 7  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
30 elpwg 4005 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
3129, 30syl 16 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ( A  e.  ~P A 
<->  A  C_  A )
)
3226, 31mpbiri 233 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  ~P A
)
3325, 32sseldd 3490 . . . 4  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  U. ( R1 " On ) )
3422, 33pm5.21ni 352 . . 3  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
3534a1d 25 . 2  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
3621, 35pm2.61i 164 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   ~Pcpw 3997   U.cuni 4234   Ord word 4867   Oncon0 4868   suc csuc 4870   dom cdm 4989   "cima 4992    Fn wfn 5573   ` cfv 5578   R1cr1 8183   rankcrnk 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186
This theorem is referenced by:  inatsk  9159
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