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Theorem rankpwg 29679
Description: The rank of a power set. Closed form of rankpw 8262. (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 4013 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21fveq2d 5870 . . 3  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
3 fveq2 5866 . . . 4  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
4 suceq 4943 . . . 4  |-  ( (
rank `  x )  =  ( rank `  A
)  ->  suc  ( rank `  x )  =  suc  ( rank `  A )
)
53, 4syl 16 . . 3  |-  ( x  =  A  ->  suc  ( rank `  x )  =  suc  ( rank `  A
) )
62, 5eqeq12d 2489 . 2  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
7 vex 3116 . . 3  |-  x  e. 
_V
87rankpw 8262 . 2  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
96, 8vtoclg 3171 1  |-  ( A  e.  V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ~Pcpw 4010   suc csuc 4880   ` cfv 5588   rankcrnk 8182
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-reg 8019  ax-inf2 8059
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077  df-r1 8183  df-rank 8184
This theorem is referenced by:  hfpw  29695
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