Theorem ranklim 4685

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` P~A) e. B))

Proof of Theorem ranklim

Step	Hyp	Ref	Expression
1		limsuc 3120	. . . 4 |- (Lim B -> ((rank` A) e. B <-> suc (rank` A) e. B))
2	1	adantl 388	. . 3 |- ((A e. V /\ Lim B) -> ((rank` A) e. B <-> suc (rank` A) e. B))
3		pweq 2403	. . . . . . . 8 |- (x = A -> P~x = P~A)
4	3	fveq2d 3728	. . . . . . 7 |- (x = A -> (rank` P~x) = (rank` P~A))
5		fveq2 3724	. . . . . . . 8 |- (x = A -> (rank` x) = (rank` A))
6		suceq 3034	. . . . . . . 8 |- ((rank` x) = (rank` A) -> suc (rank` x) = suc (rank` A))
7	5, 6	syl 10	. . . . . . 7 |- (x = A -> suc (rank` x) = suc (rank` A))
8	4, 7	eqeq12d 1489	. . . . . 6 |- (x = A -> ((rank` P~x) = suc (rank` x) <-> (rank` P~A) = suc (rank` A)))
9		visset 1813	. . . . . . 7 |- x e. V
10	9	rankpw 4684	. . . . . 6 |- (rank` P~x) = suc (rank` x)
11	8, 10	vtoclg 1847	. . . . 5 |- (A e. V -> (rank` P~A) = suc (rank` A))
12	11	eleq1d 1540	. . . 4 |- (A e. V -> ((rank` P~A) e. B <-> suc (rank` A) e. B))
13	12	adantr 389	. . 3 |- ((A e. V /\ Lim B) -> ((rank` P~A) e. B <-> suc (rank` A) e. B))
14	2, 13	bitr4d 531	. 2 |- ((A e. V /\ Lim B) -> ((rank` A) e. B <-> (rank` P~A) e. B))
15		fvprc 3721	. . . . 5 |- (-. A e. V -> (rank` A) = (/))
16		pwexb 2908	. . . . . . 7 |- (A e. V <-> P~A e. V)
17	16	negbii 187	. . . . . 6 |- (-. A e. V <-> -. P~A e. V)
18		fvprc 3721	. . . . . 6 |- (-. P~A e. V -> (rank` P~A) = (/))
19	17, 18	sylbi 199	. . . . 5 |- (-. A e. V -> (rank` P~A) = (/))
20	15, 19	eqtr4d 1510	. . . 4 |- (-. A e. V -> (rank` A) = (rank` P~A))
21	20	eleq1d 1540	. . 3 |- (-. A e. V -> ((rank` A) e. B <-> (rank` P~A) e. B))
22	21	adantr 389	. 2 |- ((-. A e. V /\ Lim B) -> ((rank` A) e. B <-> (rank` P~A) e. B))
23	14, 22	pm2.61ian 476	1 |- (Lim B -> ((rank` A) e. B <-> (rank` P~A) e. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  P~cpw 2401  Lim wlim 2949  suc csuc 2950  ` cfv 3182  rankcrnk 4642

This theorem is referenced by:  rankxplim 4712

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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