Theorem ranklim 4695

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 3126	. . . 4 |- (Lim B -> ((rank` A) e. B <-> suc (rank` A) e. B))
2	1	adantl 390	. . 3 |- ((A e. V /\ Lim B) -> ((rank` A) e. B <-> suc (rank` A) e. B))
3		pweq 2407	. . . . . . . 8 |- (x = A -> P~x = P~A)
4	3	fveq2d 3734	. . . . . . 7 |- (x = A -> (rank` P~x) = (rank` P~A))
5		fveq2 3730	. . . . . . . 8 |- (x = A -> (rank` x) = (rank` A))
6		suceq 3040	. . . . . . . 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 1492	. . . . . 6 |- (x = A -> ((rank` P~x) = suc (rank` x) <-> (rank` P~A) = suc (rank` A)))
9		visset 1816	. . . . . . 7 |- x e. V
10	9	rankpw 4694	. . . . . 6 |- (rank` P~x) = suc (rank` x)
11	8, 10	vtoclg 1850	. . . . 5 |- (A e. V -> (rank` P~A) = suc (rank` A))
12	11	eleq1d 1543	. . . 4 |- (A e. V -> ((rank` P~A) e. B <-> suc (rank` A) e. B))
13	12	adantr 391	. . 3 |- ((A e. V /\ Lim B) -> ((rank` P~A) e. B <-> suc (rank` A) e. B))
14	2, 13	bitr4d 533	. 2 |- ((A e. V /\ Lim B) -> ((rank` A) e. B <-> (rank` P~A) e. B))
15		fvprc 3727	. . . . 5 |- (-. A e. V -> (rank` A) = (/))
16		pwexb 2914	. . . . . . 7 |- (A e. V <-> P~A e. V)
17	16	negbii 187	. . . . . 6 |- (-. A e. V <-> -. P~A e. V)
18		fvprc 3727	. . . . . 6 |- (-. P~A e. V -> (rank` P~A) = (/))
19	17, 18	sylbi 199	. . . . 5 |- (-. A e. V -> (rank` P~A) = (/))
20	15, 19	eqtr4d 1513	. . . 4 |- (-. A e. V -> (rank` A) = (rank` P~A))
21	20	eleq1d 1543	. . 3 |- (-. A e. V -> ((rank` A) e. B <-> (rank` P~A) e. B))
22	21	adantr 391	. 2 |- ((-. A e. V /\ Lim B) -> ((rank` A) e. B <-> (rank` P~A) e. B))
23	14, 22	pm2.61ian 478	1 |- (Lim B -> ((rank` A) e. B <-> (rank` P~A) e. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  (/)c0 2283  P~cpw 2405  Lim wlim 2955  suc csuc 2956  ` cfv 3188  rankcrnk 4652

This theorem is referenced by:  rankxplim 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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