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Theorem rankpwi 8033
Description: The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.)
Assertion
Ref Expression
rankpwi  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )

Proof of Theorem rankpwi
StepHypRef Expression
1 rankidn 8032 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
2 rankon 8005 . . . . . . 7  |-  ( rank `  A )  e.  On
3 r1suc 7980 . . . . . . 7  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
42, 3ax-mp 5 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
54eleq2i 2507 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  ( rank `  A ) )  <->  ~P A  e.  ~P ( R1 `  ( rank `  A )
) )
6 elpwi 3872 . . . . . 6  |-  ( ~P A  e.  ~P ( R1 `  ( rank `  A
) )  ->  ~P A  C_  ( R1 `  ( rank `  A )
) )
7 pwidg 3876 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ~P A
)
8 ssel 3353 . . . . . . 7  |-  ( ~P A  C_  ( R1 `  ( rank `  A
) )  ->  ( A  e.  ~P A  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
97, 8syl5com 30 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  C_  ( R1 `  ( rank `  A ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
106, 9syl5 32 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e. 
~P ( R1 `  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
115, 10syl5bi 217 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
121, 11mtod 177 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  -.  ~P A  e.  ( R1 `  suc  ( rank `  A ) ) )
13 r1rankidb 8014 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
14 sspwb 4544 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
1513, 14sylib 196 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
1615, 4syl6sseqr 3406 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
17 fvex 5704 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
1817elpw2 4459 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
1916, 18sylibr 212 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
202onsuci 6452 . . . . 5  |-  suc  ( rank `  A )  e.  On
21 r1suc 7980 . . . . 5  |-  ( suc  ( rank `  A
)  e.  On  ->  ( R1 `  suc  suc  ( rank `  A )
)  =  ~P ( R1 `  suc  ( rank `  A ) ) )
2220, 21ax-mp 5 . . . 4  |-  ( R1
`  suc  suc  ( rank `  A ) )  =  ~P ( R1 `  suc  ( rank `  A
) )
2319, 22syl6eleqr 2534 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
24 pwwf 8017 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
25 rankr1c 8031 . . . 4  |-  ( ~P A  e.  U. ( R1 " On )  -> 
( suc  ( rank `  A )  =  (
rank `  ~P A )  <-> 
( -.  ~P A  e.  ( R1 `  suc  ( rank `  A )
)  /\  ~P A  e.  ( R1 `  suc  suc  ( rank `  A
) ) ) ) )
2624, 25sylbi 195 . . 3  |-  ( A  e.  U. ( R1
" On )  -> 
( suc  ( rank `  A )  =  (
rank `  ~P A )  <-> 
( -.  ~P A  e.  ( R1 `  suc  ( rank `  A )
)  /\  ~P A  e.  ( R1 `  suc  suc  ( rank `  A
) ) ) ) )
2712, 23, 26mpbir2and 913 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  ( rank `  ~P A ) )
2827eqcomd 2448 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3331   ~Pcpw 3863   U.cuni 4094   Oncon0 4722   suc csuc 4724   "cima 4846   ` cfv 5421   R1cr1 7972   rankcrnk 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-om 6480  df-recs 6835  df-rdg 6869  df-r1 7974  df-rank 7975
This theorem is referenced by:  rankpw  8053  r1pw  8055  r1pwcl  8057
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