Theorem ranklim 5796

Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does.

Assertion

Ref	Expression
ranklim	|- (Lim B -> ((rank` A) e. B <-> (rank` ~PA) e. B))

Proof of Theorem ranklim

Step	Hyp	Ref	Expression
1		limsuc 3933	. . . 4 |- (Lim B -> ((rank` A) e. B <-> suc (rank` A) e. B))
2	1	adantl 424	. . 3 |- ((A e. _V /\ Lim B) -> ((rank` A) e. B <-> suc (rank` A) e. B))
3		pweq 3036	. . . . . . . 8 |- (x = A -> ~Px = ~PA)
4	3	fveq2d 4685	. . . . . . 7 |- (x = A -> (rank` ~Px) = (rank` ~PA))
5		fveq2 4681	. . . . . . . 8 |- (x = A -> (rank` x) = (rank` A))
6		suceq 3729	. . . . . . . 8 |- ((rank` x) = (rank` A) -> suc (rank` x) = suc (rank` A))
7	5, 6	syl 12	. . . . . . 7 |- (x = A -> suc (rank` x) = suc (rank` A))
8	4, 7	eqeq12d 1899	. . . . . 6 |- (x = A -> ((rank` ~Px) = suc (rank` x) <-> (rank` ~PA) = suc (rank` A)))
9		visset 2295	. . . . . . 7 |- x e. _V
10	9	rankpw 5795	. . . . . 6 |- (rank` ~Px) = suc (rank` x)
11	8, 10	vtoclg 2346	. . . . 5 |- (A e. _V -> (rank` ~PA) = suc (rank` A))
12	11	eleq1d 1963	. . . 4 |- (A e. _V -> ((rank` ~PA) e. B <-> suc (rank` A) e. B))
13	12	adantr 425	. . 3 |- ((A e. _V /\ Lim B) -> ((rank` ~PA) e. B <-> suc (rank` A) e. B))
14	2, 13	bitr4d 590	. 2 |- ((A e. _V /\ Lim B) -> ((rank` A) e. B <-> (rank` ~PA) e. B))
15		fvprc 4678	. . . . 5 |- (-. A e. _V -> (rank` A) = (/))
16		pwexb 3852	. . . . . . 7 |- (A e. _V <-> ~PA e. _V)
17	16	notbii 204	. . . . . 6 |- (-. A e. _V <-> -. ~PA e. _V)
18		fvprc 4678	. . . . . 6 |- (-. ~PA e. _V -> (rank` ~PA) = (/))
19	17, 18	sylbi 216	. . . . 5 |- (-. A e. _V -> (rank` ~PA) = (/))
20	15, 19	eqtr4d 1928	. . . 4 |- (-. A e. _V -> (rank` A) = (rank` ~PA))
21	20	eleq1d 1963	. . 3 |- (-. A e. _V -> ((rank` A) e. B <-> (rank` ~PA) e. B))
22	21	adantr 425	. 2 |- ((-. A e. _V /\ Lim B) -> ((rank` A) e. B <-> (rank` ~PA) e. B))
23	14, 22	pm2.61ian 534	1 |- (Lim B -> ((rank` A) e. B <-> (rank` ~PA) e. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ~Pcpw 3032  Lim wlim 3658  suc csuc 3659  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  rankxplim 5823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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