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Theorem r1pw 7727
Description: A stronger property of  R1 than rankpw 7725. 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 7705 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
21eleq1d 2470 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  ~P A )  e.  suc  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
3 eloni 4551 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
4 ordsucelsuc 4761 . . . . . . 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 193 . . . . 5  |-  ( B  e.  On  ->  ( suc  ( rank `  A
)  e.  suc  B  <->  (
rank `  A )  e.  B ) )
72, 6sylan9bb 681 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ( rank `  ~P A )  e.  suc  B  <-> 
( rank `  A )  e.  B ) )
8 pwwf 7689 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
98biimpi 187 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  U. ( R1 " On ) )
10 suceloni 4752 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
11 r1fnon 7649 . . . . . . 7  |-  R1  Fn  On
12 fndm 5503 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 8 . . . . . 6  |-  dom  R1  =  On
1410, 13syl6eleqr 2495 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  dom  R1 )
15 rankr1ag 7684 . . . . 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 464 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( ~P A  e.  ( R1 `  suc  B )  <->  ( rank `  ~P A )  e.  suc  B ) )
1713eleq2i 2468 . . . . 5  |-  ( B  e.  dom  R1  <->  B  e.  On )
18 rankr1ag 7684 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
1917, 18sylan2br 463 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <-> 
( rank `  A )  e.  B ) )
207, 16, 193bitr4rd 278 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  On )  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
2120ex 424 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
22 r1elwf 7678 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
23 r1elwf 7678 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  U. ( R1 " On ) )
24 r1elssi 7687 . . . . . 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 3327 . . . . . 6  |-  A  C_  A
27 elex 2924 . . . . . . . 8  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
28 pwexb 4712 . . . . . . . 8  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2927, 28sylibr 204 . . . . . . 7  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
30 elpwg 3766 . . . . . . 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 225 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  ~P A
)
3325, 32sseldd 3309 . . . 4  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  U. ( R1 " On ) )
3422, 33pm5.21ni 342 . . 3  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
3534a1d 23 . 2  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
3621, 35pm2.61i 158 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837   "cima 4840    Fn wfn 5408   ` cfv 5413   R1cr1 7644   rankcrnk 7645
This theorem is referenced by:  inatsk  8609
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647
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