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Theorem rankpwi 8232
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 8231 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
2 rankon 8204 . . . . . . 7  |-  ( rank `  A )  e.  On
3 r1suc 8179 . . . . . . 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 2532 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  ( rank `  A ) )  <->  ~P A  e.  ~P ( R1 `  ( rank `  A )
) )
6 elpwi 4008 . . . . . 6  |-  ( ~P A  e.  ~P ( R1 `  ( rank `  A
) )  ->  ~P A  C_  ( R1 `  ( rank `  A )
) )
7 pwidg 4012 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ~P A
)
8 ssel 3483 . . . . . . 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 8213 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
14 sspwb 4686 . . . . . . 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 3536 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
17 fvex 5858 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
1817elpw2 4601 . . . . 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 6646 . . . . 5  |-  suc  ( rank `  A )  e.  On
21 r1suc 8179 . . . . 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 2553 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
24 pwwf 8216 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
25 rankr1c 8230 . . . 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 920 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  ( rank `  ~P A ) )
2827eqcomd 2462 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   Oncon0 4867   suc csuc 4869   "cima 4991   ` cfv 5570   R1cr1 8171   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by:  rankpw  8252  r1pw  8254  r1pwcl  8256
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