MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankpwi Structured version   Unicode version

Theorem rankpwi 8253
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 8252 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
2 rankon 8225 . . . . . . 7  |-  ( rank `  A )  e.  On
3 r1suc 8200 . . . . . . 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 2545 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  ( rank `  A ) )  <->  ~P A  e.  ~P ( R1 `  ( rank `  A )
) )
6 elpwi 4025 . . . . . 6  |-  ( ~P A  e.  ~P ( R1 `  ( rank `  A
) )  ->  ~P A  C_  ( R1 `  ( rank `  A )
) )
7 pwidg 4029 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ~P A
)
8 ssel 3503 . . . . . . 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 8234 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
14 sspwb 4702 . . . . . . 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 3556 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
17 fvex 5882 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
1817elpw2 4617 . . . . 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 6668 . . . . 5  |-  suc  ( rank `  A )  e.  On
21 r1suc 8200 . . . . 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 2566 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
24 pwwf 8237 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
25 rankr1c 8251 . . . 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 2475 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 1379    e. wcel 1767    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   Oncon0 4884   suc csuc 4886   "cima 5008   ` cfv 5594   R1cr1 8192   rankcrnk 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-rank 8195
This theorem is referenced by:  rankpw  8273  r1pw  8275  r1pwcl  8277
  Copyright terms: Public domain W3C validator