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Theorem r1pw 7771
Description: A stronger property of  R1 than rankpw 7769. 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 7749 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
21eleq1d 2502 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  ~P A )  e.  suc  B  <->  suc  ( rank `  A
)  e.  suc  B
) )
3 eloni 4591 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
4 ordsucelsuc 4802 . . . . . . 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 7733 . . . . . 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 4793 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
11 r1fnon 7693 . . . . . . 7  |-  R1  Fn  On
12 fndm 5544 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 8 . . . . . 6  |-  dom  R1  =  On
1410, 13syl6eleqr 2527 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  dom  R1 )
15 rankr1ag 7728 . . . . 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 2500 . . . . 5  |-  ( B  e.  dom  R1  <->  B  e.  On )
18 rankr1ag 7728 . . . . 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 7722 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
23 r1elwf 7722 . . . . . 6  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  U. ( R1 " On ) )
24 r1elssi 7731 . . . . . 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 3367 . . . . . 6  |-  A  C_  A
27 elex 2964 . . . . . . . 8  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
28 pwexb 4753 . . . . . . . 8  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2927, 28sylibr 204 . . . . . . 7  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
30 elpwg 3806 . . . . . . 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 3349 . . . 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 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   Ord word 4580   Oncon0 4581   suc csuc 4583   dom cdm 4878   "cima 4881    Fn wfn 5449   ` cfv 5454   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  inatsk  8653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-rank 7691
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