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Theorem ranklim 7705
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 4771 . . . 4  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
21adantl 453 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
3 pweq 3747 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
43fveq2d 5674 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
5 fveq2 5670 . . . . . . . 8  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
6 suceq 4589 . . . . . . . 8  |-  ( (
rank `  x )  =  ( rank `  A
)  ->  suc  ( rank `  x )  =  suc  ( rank `  A )
)
75, 6syl 16 . . . . . . 7  |-  ( x  =  A  ->  suc  ( rank `  x )  =  suc  ( rank `  A
) )
84, 7eqeq12d 2403 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
9 vex 2904 . . . . . . 7  |-  x  e. 
_V
109rankpw 7704 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
118, 10vtoclg 2956 . . . . 5  |-  ( A  e.  _V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
1211eleq1d 2455 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
1312adantr 452 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
142, 13bitr4d 248 . 2  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
15 fvprc 5664 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
16 pwexb 4695 . . . . . 6  |-  ( A  e.  _V  <->  ~P A  e.  _V )
17 fvprc 5664 . . . . . 6  |-  ( -. 
~P A  e.  _V  ->  ( rank `  ~P A )  =  (/) )
1816, 17sylnbi 298 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  ~P A )  =  (/) )
1915, 18eqtr4d 2424 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  ( rank `  ~P A ) )
2019eleq1d 2455 . . 3  |-  ( -.  A  e.  _V  ->  ( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2120adantr 452 . 2  |-  ( ( -.  A  e.  _V  /\ 
Lim  B )  -> 
( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2214, 21pm2.61ian 766 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573   ~Pcpw 3744   Lim wlim 4525   suc csuc 4526   ` cfv 5396   rankcrnk 7624
This theorem is referenced by:  rankxplim  7738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-reg 7495  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-recs 6571  df-rdg 6606  df-r1 7625  df-rank 7626
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