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Theorem r1pw 8164
Description: A stronger property of  R1 than rankpw 8162. 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 8142 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
21eleq1d 2523 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  ~P A )  e.  suc  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
3 eloni 4838 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
4 ordsucelsuc 6544 . . . . . . 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 8126 . . . . . 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 6535 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
11 r1fnon 8086 . . . . . . 7  |-  R1  Fn  On
12 fndm 5619 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 5 . . . . . 6  |-  dom  R1  =  On
1410, 13syl6eleqr 2553 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  dom  R1 )
15 rankr1ag 8121 . . . . 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 2532 . . . . 5  |-  ( B  e.  dom  R1  <->  B  e.  On )
18 rankr1ag 8121 . . . . 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 8115 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
23 r1elwf 8115 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  U. ( R1 " On ) )
24 r1elssi 8124 . . . . . 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 3484 . . . . . 6  |-  A  C_  A
27 elex 3087 . . . . . . . 8  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
28 pwexb 6498 . . . . . . . 8  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2927, 28sylibr 212 . . . . . . 7  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
30 elpwg 3977 . . . . . . 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 3466 . . . 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 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437   ~Pcpw 3969   U.cuni 4200   Ord word 4827   Oncon0 4828   suc csuc 4830   dom cdm 4949   "cima 4952    Fn wfn 5522   ` cfv 5527   R1cr1 8081   rankcrnk 8082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-r1 8083  df-rank 8084
This theorem is referenced by:  inatsk  9057
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