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Theorem ranklim 8163
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 6571 . . . 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 3972 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
43fveq2d 5804 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
5 fveq2 5800 . . . . . . . 8  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
6 suceq 4893 . . . . . . . 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 2476 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
9 vex 3081 . . . . . . 7  |-  x  e. 
_V
109rankpw 8162 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
118, 10vtoclg 3136 . . . . 5  |-  ( A  e.  _V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
1211eleq1d 2523 . . . 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 5794 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
16 pwexb 6498 . . . . . 6  |-  ( A  e.  _V  <->  ~P A  e.  _V )
17 fvprc 5794 . . . . . 6  |-  ( -. 
~P A  e.  _V  ->  ( rank `  ~P A )  =  (/) )
1816, 17sylnbi 306 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  ~P A )  =  (/) )
1915, 18eqtr4d 2498 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  ( rank `  ~P A ) )
2019eleq1d 2523 . . 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 788 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 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746   ~Pcpw 3969   Lim wlim 4829   suc csuc 4830   ` cfv 5527   rankcrnk 8082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-reg 7919  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-r1 8083  df-rank 8084
This theorem is referenced by:  rankxplim  8198
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