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.

Step	Hyp	Ref	Expression
1		limsuc 6655	. . . 4 |- ( Lim B -> ( ( rank ` A ) e. B <-> suc ( rank ` A ) e. B ) )
2	1	adantl 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 )
4	3	fveq2d 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 ) )
7	5, 6	syl 16	. . . . . . 7 |- ( x = A -> suc ( rank ` x ) = suc ( rank ` A ) )
8	4, 7	eqeq12d 2482	. . . . . 6 |- ( x = A -> ( ( rank ` ~P x ) = suc ( rank ` x ) <-> ( rank ` ~P A ) = suc ( rank ` A ) ) )
9		vex 3109	. . . . . . 7 |- x e. _V
10	9	rankpw 8250	. . . . . 6 |- ( rank ` ~P x ) = suc ( rank ` x )
11	8, 10	vtoclg 3164	. . . . 5 |- ( A e. _V -> ( rank ` ~P A ) = suc ( rank ` A ) )
12	11	eleq1d 2529	. . . 4 |- ( A e. _V -> ( ( rank ` ~P A ) e. B <-> suc ( rank ` A ) e. B ) )
13	12	adantr 465	. . 3 |- ( ( A e. _V /\ Lim B ) -> ( ( rank ` ~P A ) e. B <-> suc ( rank ` A ) e. B ) )
14	2, 13	bitr4d 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 ) = (/) )
18	16, 17	sylnbi 306	. . . . 5 |- ( -. A e. _V -> ( rank ` ~P A ) = (/) )
19	15, 18	eqtr4d 2504	. . . 4 |- ( -. A e. _V -> ( rank ` A ) = ( rank ` ~P A ) )
20	19	eleq1d 2529	. . 3 |- ( -. A e. _V -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )
21	20	adantr 465	. 2 |- ( ( -. A e. _V /\ Lim B ) -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )
22	14, 21	pm2.61ian 788	1 |- ( Lim B -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )

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