Theorem r1pwcl 4659

Description: The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.)

Assertion

Ref	Expression
r1pwcl	|- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))

Proof of Theorem r1pwcl

Step	Hyp	Ref	Expression
1		r1lim 4625	. . . . . . 7 |- ((B e. V /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
2	1	eleq2d 1533	. . . . . 6 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> A e. U_x e. B (R1` x)))
3		eliun 2560	. . . . . 6 |- (A e. U_x e. B (R1` x) <-> E.x e. B A e. (R1` x))
4	2, 3	syl6bb 534	. . . . 5 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
5		onelon 2962	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6		limelon 3022	. . . . . . . 8 |- ((B e. V /\ Lim B) -> B e. On)
7	5, 6	sylan 448	. . . . . . 7 |- (((B e. V /\ Lim B) /\ x e. B) -> x e. On)
8		r1pw 4658	. . . . . . 7 |- (x e. On -> (A e. (R1` x) <-> P~A e. (R1` suc x)))
9	7, 8	syl 10	. . . . . 6 |- (((B e. V /\ Lim B) /\ x e. B) -> (A e. (R1` x) <-> P~A e. (R1` suc x)))
10	9	rexbidva 1652	. . . . 5 |- ((B e. V /\ Lim B) -> (E.x e. B A e. (R1` x) <-> E.x e. B P~A e. (R1` suc x)))
11		limsuc 3110	. . . . . . . . . . . 12 |- (Lim B -> (x e. B <-> suc x e. B))
12	11	anbi1d 615	. . . . . . . . . . 11 |- (Lim B -> ((x e. B /\ P~A e. (R1` suc x)) <-> (suc x e. B /\ P~A e. (R1` suc x))))
13		visset 1804	. . . . . . . . . . . . 13 |- x e. V
14	13	sucex 3040	. . . . . . . . . . . 12 |- suc x e. V
15		eleq1 1526	. . . . . . . . . . . . 13 |- (y = suc x -> (y e. B <-> suc x e. B))
16		fveq2 3709	. . . . . . . . . . . . . 14 |- (y = suc x -> (R1` y) = (R1` suc x))
17	16	eleq2d 1533	. . . . . . . . . . . . 13 |- (y = suc x -> (P~A e. (R1` y) <-> P~A e. (R1` suc x)))
18	15, 17	anbi12d 626	. . . . . . . . . . . 12 |- (y = suc x -> ((y e. B /\ P~A e. (R1` y)) <-> (suc x e. B /\ P~A e. (R1` suc x))))
19	14, 18	cla4ev 1860	. . . . . . . . . . 11 |- ((suc x e. B /\ P~A e. (R1` suc x)) -> E.y(y e. B /\ P~A e. (R1` y)))
20	12, 19	syl6bi 214	. . . . . . . . . 10 |- (Lim B -> ((x e. B /\ P~A e. (R1` suc x)) -> E.y(y e. B /\ P~A e. (R1` y))))
21	20	19.23adv 1209	. . . . . . . . 9 |- (Lim B -> (E.x(x e. B /\ P~A e. (R1` suc x)) -> E.y(y e. B /\ P~A e. (R1` y))))
22		df-rex 1642	. . . . . . . . 9 |- (E.x e. B P~A e. (R1` suc x) <-> E.x(x e. B /\ P~A e. (R1` suc x)))
23		df-rex 1642	. . . . . . . . 9 |- (E.y e. B P~A e. (R1` y) <-> E.y(y e. B /\ P~A e. (R1` y)))
24	21, 22, 23	3imtr4g 551	. . . . . . . 8 |- (Lim B -> (E.x e. B P~A e. (R1` suc x) -> E.y e. B P~A e. (R1` y)))
25		fveq2 3709	. . . . . . . . . 10 |- (x = y -> (R1` x) = (R1` y))
26	25	eleq2d 1533	. . . . . . . . 9 |- (x = y -> (P~A e. (R1` x) <-> P~A e. (R1` y)))
27	26	cbvrexv 1792	. . . . . . . 8 |- (E.x e. B P~A e. (R1` x) <-> E.y e. B P~A e. (R1` y))
28	24, 27	syl6ibr 213	. . . . . . 7 |- (Lim B -> (E.x e. B P~A e. (R1` suc x) -> E.x e. B P~A e. (R1` x)))
29	28	adantl 388	. . . . . 6 |- ((B e. V /\ Lim B) -> (E.x e. B P~A e. (R1` suc x) -> E.x e. B P~A e. (R1` x)))
30	7	ex 373	. . . . . . . 8 |- ((B e. V /\ Lim B) -> (x e. B -> x e. On))
31		sssucid 3037	. . . . . . . . . . . 12 |- x (_ suc x
32		r1ord3 4629	. . . . . . . . . . . 12 |- ((x e. On /\ suc x e. On) -> (x (_ suc x -> (R1` x) (_ (R1` suc x)))
33	31, 32	mpi 44	. . . . . . . . . . 11 |- ((x e. On /\ suc x e. On) -> (R1` x) (_ (R1` suc x))
34		sucelon 3058	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
35	33, 34	sylan2b 452	. . . . . . . . . 10 |- ((x e. On /\ x e. On) -> (R1` x) (_ (R1` suc x))
36	35	anidms 434	. . . . . . . . 9 |- (x e. On -> (R1` x) (_ (R1` suc x))
37	36	sseld 2057	. . . . . . . 8 |- (x e. On -> (P~A e. (R1` x) -> P~A e. (R1` suc x)))
38	30, 37	syl6 22	. . . . . . 7 |- ((B e. V /\ Lim B) -> (x e. B -> (P~A e. (R1` x) -> P~A e. (R1` suc x))))
39	38	r19.22dv 1729	. . . . . 6 |- ((B e. V /\ Lim B) -> (E.x e. B P~A e. (R1` x) -> E.x e. B P~A e. (R1` suc x)))
40	29, 39	impbid 514	. . . . 5 |- ((B e. V /\ Lim B) -> (E.x e. B P~A e. (R1` suc x) <-> E.x e. B P~A e. (R1` x)))
41	4, 10, 40	3bitrd 542	. . . 4 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
42	1	eleq2d 1533	. . . . 5 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> P~A e. U_x e. B (R1` x)))
43		eliun 2560	. . . . 5 |- (P~A e. U_x e. B (R1` x) <-> E.x e. B P~A e. (R1` x))
44	42, 43	syl6bb 534	. . . 4 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
45	41, 44	bitr4d 529	. . 3 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> P~A e. (R1` B)))
46	45	ex 373	. 2 |- (B e. V -> (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B))))
47		n0i 2275	. . . . 5 |- (A e. (R1` B) -> -. (R1` B) = (/))
48		fvprc 3706	. . . . 5 |- (-. B e. V -> (R1` B) = (/))
49	47, 48	nsyl2 118	. . . 4 |- (A e. (R1` B) -> B e. V)
50		n0i 2275	. . . . 5 |- (P~A e. (R1` B) -> -. (R1` B) = (/))
51	50, 48	nsyl2 118	. . . 4 |- (P~A e. (R1` B) -> B e. V)
52	49, 51	pm5.21ni 676	. . 3 |- (-. B e. V -> (A e. (R1` B) <-> P~A e. (R1` B)))
53	52	a1d 12	. 2 |- (-. B e. V -> (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B))))
54	46, 53	pm2.61i 126	1 |- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E.wrex 1638  Vcvv 1802   (_ wss 2037  (/)c0 2270  P~cpw 2391  U_ciun 2556  Oncon0 2938  Lim wlim 2939  suc csuc 2940  ` cfv 3172  R1cr1 4613

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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