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Theorem r1pw 8268
Description: A stronger property of  R1 than rankpw 8266. 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 8246 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
21eleq1d 2490 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  ~P A )  e.  suc  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
3 eloni 5395 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
4 ordsucelsuc 6607 . . . . . . 7  |-  ( Ord 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
53, 4syl 17 . . . . . 6  |-  ( B  e.  On  ->  (
( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
65bicomd 204 . . . . 5  |-  ( B  e.  On  ->  ( suc  ( rank `  A
)  e.  suc  B  <->  (
rank `  A )  e.  B ) )
72, 6sylan9bb 704 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ( rank `  ~P A )  e.  suc  B  <-> 
( rank `  A )  e.  B ) )
8 pwwf 8230 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
98biimpi 197 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  U. ( R1 " On ) )
10 suceloni 6598 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
11 r1fnon 8190 . . . . . . 7  |-  R1  Fn  On
12 fndm 5636 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 5 . . . . . 6  |-  dom  R1  =  On
1410, 13syl6eleqr 2517 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  dom  R1 )
15 rankr1ag 8225 . . . . 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 479 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ~P A  e.  ( R1 `  suc  B )  <->  ( rank `  ~P A )  e.  suc  B ) )
1713eleq2i 2498 . . . . 5  |-  ( B  e.  dom  R1  <->  B  e.  On )
18 rankr1ag 8225 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
1917, 18sylan2br 478 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <-> 
( rank `  A )  e.  B ) )
207, 16, 193bitr4rd 289 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
2120ex 435 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
22 r1elwf 8219 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
23 r1elwf 8219 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  U. ( R1 " On ) )
24 r1elssi 8228 . . . . . 6  |-  ( ~P A  e.  U. ( R1 " On )  ->  ~P A  C_  U. ( R1 " On ) )
2523, 24syl 17 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  C_  U. ( R1 " On ) )
26 ssid 3426 . . . . . 6  |-  A  C_  A
27 elex 3031 . . . . . . . 8  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
28 pwexb 6560 . . . . . . . 8  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2927, 28sylibr 215 . . . . . . 7  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
30 elpwg 3932 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
3129, 30syl 17 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ( A  e.  ~P A 
<->  A  C_  A )
)
3226, 31mpbiri 236 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  ~P A
)
3325, 32sseldd 3408 . . . 4  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  U. ( R1 " On ) )
3422, 33pm5.21ni 353 . . 3  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
3534a1d 26 . 2  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
3621, 35pm2.61i 167 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    C_ wss 3379   ~Pcpw 3924   U.cuni 4162   dom cdm 4796   "cima 4799   Ord word 5384   Oncon0 5385   suc csuc 5387    Fn wfn 5539   ` cfv 5544   R1cr1 8185   rankcrnk 8186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-r1 8187  df-rank 8188
This theorem is referenced by:  inatsk  9154
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