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Theorem ranklim 7516
Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)
Assertion
Ref Expression
ranklim  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )

Proof of Theorem ranklim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 limsuc 4640 . . . 4  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
21adantl 452 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
3 pweq 3628 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
43fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
5 fveq2 5525 . . . . . . . 8  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
6 suceq 4457 . . . . . . . 8  |-  ( (
rank `  x )  =  ( rank `  A
)  ->  suc  ( rank `  x )  =  suc  ( rank `  A )
)
75, 6syl 15 . . . . . . 7  |-  ( x  =  A  ->  suc  ( rank `  x )  =  suc  ( rank `  A
) )
84, 7eqeq12d 2297 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
9 vex 2791 . . . . . . 7  |-  x  e. 
_V
109rankpw 7515 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
118, 10vtoclg 2843 . . . . 5  |-  ( A  e.  _V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
1211eleq1d 2349 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
1312adantr 451 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
142, 13bitr4d 247 . 2  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
15 fvprc 5519 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
16 pwexb 4564 . . . . . 6  |-  ( A  e.  _V  <->  ~P A  e.  _V )
17 fvprc 5519 . . . . . 6  |-  ( -. 
~P A  e.  _V  ->  ( rank `  ~P A )  =  (/) )
1816, 17sylnbi 297 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  ~P A )  =  (/) )
1915, 18eqtr4d 2318 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  ( rank `  ~P A ) )
2019eleq1d 2349 . . 3  |-  ( -.  A  e.  _V  ->  ( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2120adantr 451 . 2  |-  ( ( -.  A  e.  _V  /\ 
Lim  B )  -> 
( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2214, 21pm2.61ian 765 1  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   Lim wlim 4393   suc csuc 4394   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  rankxplim  7549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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