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Theorem rankpw 8252
Description: The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
rankpw.1  |-  A  e. 
_V
Assertion
Ref Expression
rankpw  |-  ( rank `  ~P A )  =  suc  ( rank `  A
)

Proof of Theorem rankpw
StepHypRef Expression
1 rankpw.1 . . 3  |-  A  e. 
_V
2 unir1 8222 . . 3  |-  U. ( R1 " On )  =  _V
31, 2eleqtrri 2541 . 2  |-  A  e. 
U. ( R1 " On )
4 rankpwi 8232 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  ~P A )  =  suc  ( rank `  A ) )
53, 4ax-mp 5 1  |-  ( rank `  ~P A )  =  suc  ( rank `  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   _Vcvv 3106   ~Pcpw 3999   U.cuni 4235   Oncon0 4867   suc csuc 4869   "cima 4991   ` cfv 5570   R1cr1 8171   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by:  ranklim  8253  r1pwALT  8255  rankuni  8272  rankc2  8280  rankxpu  8285  rankmapu  8287  rankpwg  30057
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