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Theorem rankpwi 7582
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 7581 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
2 rankon 7554 . . . . . . 7  |-  ( rank `  A )  e.  On
3 r1suc 7529 . . . . . . 7  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
42, 3ax-mp 8 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
54eleq2i 2422 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  ( rank `  A ) )  <->  ~P A  e.  ~P ( R1 `  ( rank `  A )
) )
6 elpwi 3709 . . . . . 6  |-  ( ~P A  e.  ~P ( R1 `  ( rank `  A
) )  ->  ~P A  C_  ( R1 `  ( rank `  A )
) )
7 pwidg 3713 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ~P A
)
8 ssel 3250 . . . . . . 7  |-  ( ~P A  C_  ( R1 `  ( rank `  A
) )  ->  ( A  e.  ~P A  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
97, 8syl5com 26 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  C_  ( R1 `  ( rank `  A ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
106, 9syl5 28 . . . . 5  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e. 
~P ( R1 `  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
115, 10syl5bi 208 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( ~P A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
121, 11mtod 168 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  -.  ~P A  e.  ( R1 `  suc  ( rank `  A ) ) )
13 r1rankidb 7563 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
14 sspwb 4302 . . . . . . 7  |-  ( A 
C_  ( R1 `  ( rank `  A )
)  <->  ~P A  C_  ~P ( R1 `  ( rank `  A ) ) )
1513, 14sylib 188 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ~P ( R1 `  ( rank `  A
) ) )
1615, 4syl6sseqr 3301 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  C_  ( R1
`  suc  ( rank `  A ) ) )
17 fvex 5619 . . . . . 6  |-  ( R1
`  suc  ( rank `  A ) )  e. 
_V
1817elpw2 4254 . . . . 5  |-  ( ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) )  <->  ~P A  C_  ( R1 `  suc  ( rank `  A )
) )
1916, 18sylibr 203 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ~P ( R1 `  suc  ( rank `  A ) ) )
202onsuci 4708 . . . . 5  |-  suc  ( rank `  A )  e.  On
21 r1suc 7529 . . . . 5  |-  ( suc  ( rank `  A
)  e.  On  ->  ( R1 `  suc  suc  ( rank `  A )
)  =  ~P ( R1 `  suc  ( rank `  A ) ) )
2220, 21ax-mp 8 . . . 4  |-  ( R1
`  suc  suc  ( rank `  A ) )  =  ~P ( R1 `  suc  ( rank `  A
) )
2319, 22syl6eleqr 2449 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  ~P A  e.  ( R1 `  suc  suc  ( rank `  A ) ) )
24 pwwf 7566 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  ~P A  e.  U. ( R1 " On ) )
25 rankr1c 7580 . . . 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 187 . . 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 888 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  ( rank `  ~P A ) )
2827eqcomd 2363 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   ~Pcpw 3701   U.cuni 3906   Oncon0 4471   suc csuc 4473   "cima 4771   ` cfv 5334   R1cr1 7521   rankcrnk 7522
This theorem is referenced by:  rankpw  7602  r1pw  7604  r1pwcl  7606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-recs 6472  df-rdg 6507  df-r1 7523  df-rank 7524
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