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Theorem rankpwg 30747
Description: The rank of a power set. Closed form of rankpw 8304. (Contributed by Scott Fenton, 16-Jul-2015.)
Assertion
Ref Expression
rankpwg  |-  ( A  e.  V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )

Proof of Theorem rankpwg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3979 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21fveq2d 5876 . . 3  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
3 fveq2 5872 . . . 4  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
4 suceq 5498 . . . 4  |-  ( (
rank `  x )  =  ( rank `  A
)  ->  suc  ( rank `  x )  =  suc  ( rank `  A )
)
53, 4syl 17 . . 3  |-  ( x  =  A  ->  suc  ( rank `  x )  =  suc  ( rank `  A
) )
62, 5eqeq12d 2442 . 2  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
7 vex 3081 . . 3  |-  x  e. 
_V
87rankpw 8304 . 2  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
96, 8vtoclg 3136 1  |-  ( A  e.  V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   ~Pcpw 3976   suc csuc 5435   ` cfv 5592   rankcrnk 8224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-reg 8098  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-r1 8225  df-rank 8226
This theorem is referenced by:  hfpw  30763
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