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.

Step	Hyp	Ref	Expression
1		limsuc 6683	. . . 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 4018	. . . . . . . 8 |- ( x = A -> ~P x = ~P A )
4	3	fveq2d 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 ) )
7	5, 6	syl 16	. . . . . . 7 |- ( x = A -> suc ( rank ` x ) = suc ( rank ` A ) )
8	4, 7	eqeq12d 2479	. . . . . 6 |- ( x = A -> ( ( rank ` ~P x ) = suc ( rank ` x ) <-> ( rank ` ~P A ) = suc ( rank ` A ) ) )
9		vex 3112	. . . . . . 7 |- x e. _V
10	9	rankpw 8278	. . . . . 6 |- ( rank ` ~P x ) = suc ( rank ` x )
11	8, 10	vtoclg 3167	. . . . 5 |- ( A e. _V -> ( rank ` ~P A ) = suc ( rank ` A ) )
12	11	eleq1d 2526	. . . 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 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 ) = (/) )
18	16, 17	sylnbi 306	. . . . 5 |- ( -. A e. _V -> ( rank ` ~P A ) = (/) )
19	15, 18	eqtr4d 2501	. . . 4 |- ( -. A e. _V -> ( rank ` A ) = ( rank ` ~P A ) )
20	19	eleq1d 2526	. . 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 790	1 |- ( Lim B -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )

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