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Theorem ranklim 8251
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 6655 . . . 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 4006 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
43fveq2d 5861 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  ~P x )  =  ( rank `  ~P A ) )
5 fveq2 5857 . . . . . . . 8  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
6 suceq 4936 . . . . . . . 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 2482 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  ~P x
)  =  suc  ( rank `  x )  <->  ( rank `  ~P A )  =  suc  ( rank `  A
) ) )
9 vex 3109 . . . . . . 7  |-  x  e. 
_V
109rankpw 8250 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
118, 10vtoclg 3164 . . . . 5  |-  ( A  e.  _V  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
1211eleq1d 2529 . . . 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 5851 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
16 pwexb 6582 . . . . . 6  |-  ( A  e.  _V  <->  ~P A  e.  _V )
17 fvprc 5851 . . . . . 6  |-  ( -. 
~P A  e.  _V  ->  ( rank `  ~P A )  =  (/) )
1816, 17sylnbi 306 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  ~P A )  =  (/) )
1915, 18eqtr4d 2504 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  ( rank `  ~P A ) )
2019eleq1d 2529 . . 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 1374    e. wcel 1762   _Vcvv 3106   (/)c0 3778   ~Pcpw 4003   Lim wlim 4872   suc csuc 4873   ` cfv 5579   rankcrnk 8170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066  df-r1 8171  df-rank 8172
This theorem is referenced by:  rankxplim  8286
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