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Theorem rankpwi 7463
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 7462 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
2 rankon 7435 . . . . . . 7  |-  ( rank `  A )  e.  On
3 r1suc 7410 . . . . . . 7  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
42, 3ax-mp 10 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
54eleq2i 2322 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  ( rank `  A ) )  <->  ~P A  e.  ~P ( R1 `  ( rank `  A )
) )
6 elpwi 3607 . . . . . 6  |-  ( ~P A  e.  ~P ( R1 `  ( rank `  A
) )  ->  ~P A  C_  ( R1 `  ( rank `  A )
) )
7 pwidg 3611 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ~P A
)
8 ssel 3149 . . . . . . 7  |-  ( ~P A  C_  ( R1 `  ( rank `  A
) )  ->  ( A  e.  ~P A  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
97, 8syl5com 28 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  C_  ( R1 `  ( rank `  A ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
106, 9syl5 30 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e. 
~P ( R1 `  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
115, 10syl5bi 210 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
121, 11mtod 170 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  -.  ~P A  e.  ( R1 `  suc  ( rank `  A ) ) )
13 r1rankidb 7444 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
14 sspwb 4195 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
1513, 14sylib 190 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
1615, 4syl6sseqr 3200 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
17 fvex 5472 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
1817elpw2 4142 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
1916, 18sylibr 205 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
202onsuci 4601 . . . . 5  |-  suc  ( rank `  A )  e.  On
21 r1suc 7410 . . . . 5  |-  ( suc  ( rank `  A
)  e.  On  ->  ( R1 `  suc  suc  ( rank `  A )
)  =  ~P ( R1 `  suc  ( rank `  A ) ) )
2220, 21ax-mp 10 . . . 4  |-  ( R1
`  suc  suc  ( rank `  A ) )  =  ~P ( R1 `  suc  ( rank `  A
) )
2319, 22syl6eleqr 2349 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
24 pwwf 7447 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
25 rankr1c 7461 . . . 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 189 . . 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 893 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  ( rank `  ~P A ) )
2827eqcomd 2263 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3127   ~Pcpw 3599   U.cuni 3801   Oncon0 4364   suc csuc 4366   "cima 4664   ` cfv 4673   R1cr1 7402   rankcrnk 7403
This theorem is referenced by:  rankpw  7483  r1pw  7485  r1pwcl  7487
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404  df-rank 7405
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