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Theorem ranklim 8279
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 6683 . . . 4  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
21adantl 466 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  suc  ( rank `  A
)  e.  B ) )
3 pweq 4018 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
43fveq2d 5876 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
5 fveq2 5872 . . . . . . . 8  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
6 suceq 4952 . . . . . . . 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 2479 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
9 vex 3112 . . . . . . 7  |-  x  e. 
_V
109rankpw 8278 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
118, 10vtoclg 3167 . . . . 5  |-  ( A  e.  _V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
1211eleq1d 2526 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
1312adantr 465 . . 3  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  ~P A )  e.  B  <->  suc  ( rank `  A )  e.  B
) )
142, 13bitr4d 256 . 2  |-  ( ( A  e.  _V  /\  Lim  B )  ->  (
( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
15 fvprc 5866 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
16 pwexb 6610 . . . . . 6  |-  ( A  e.  _V  <->  ~P A  e.  _V )
17 fvprc 5866 . . . . . 6  |-  ( -. 
~P A  e.  _V  ->  ( rank `  ~P A )  =  (/) )
1816, 17sylnbi 306 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  ~P A )  =  (/) )
1915, 18eqtr4d 2501 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  ( rank `  ~P A ) )
2019eleq1d 2526 . . 3  |-  ( -.  A  e.  _V  ->  ( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2120adantr 465 . 2  |-  ( ( -.  A  e.  _V  /\ 
Lim  B )  -> 
( ( rank `  A
)  e.  B  <->  ( rank `  ~P A )  e.  B ) )
2214, 21pm2.61ian 790 1  |-  ( Lim 
B  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ~Pcpw 4015   Lim wlim 4888   suc csuc 4889   ` cfv 5594   rankcrnk 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200
This theorem is referenced by:  rankxplim  8314
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