Theorem r1pwcl 4697

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 4663	. . . . . . 7 |- ((B e. V /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
2	1	eleq2d 1544	. . . . . 6 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> A e. U_x e. B (R1` x)))
3		eliun 2574	. . . . . 6 |- (A e. U_x e. B (R1` x) <-> E.x e. B A e. (R1` x))
4	2, 3	syl6bb 538	. . . . 5 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
5		onelon 2978	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6		limelon 3038	. . . . . . . 8 |- ((B e. V /\ Lim B) -> B e. On)
7	5, 6	sylan 450	. . . . . . 7 |- (((B e. V /\ Lim B) /\ x e. B) -> x e. On)
8		r1pw 4696	. . . . . . 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 1663	. . . . 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 3126	. . . . . . . . . . . 12 |- (Lim B -> (x e. B <-> suc x e. B))
12	11	anbi1d 619	. . . . . . . . . . 11 |- (Lim B -> ((x e. B /\ P~A e. (R1` suc x)) <-> (suc x e. B /\ P~A e. (R1` suc x))))
13		visset 1816	. . . . . . . . . . . . 13 |- x e. V
14	13	sucex 3056	. . . . . . . . . . . 12 |- suc x e. V
15		eleq1 1537	. . . . . . . . . . . . 13 |- (y = suc x -> (y e. B <-> suc x e. B))
16		fveq2 3730	. . . . . . . . . . . . . 14 |- (y = suc x -> (R1` y) = (R1` suc x))
17	16	eleq2d 1544	. . . . . . . . . . . . 13 |- (y = suc x -> (P~A e. (R1` y) <-> P~A e. (R1` suc x)))
18	15, 17	anbi12d 630	. . . . . . . . . . . 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 1872	. . . . . . . . . . 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 1216	. . . . . . . . 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 1653	. . . . . . . . 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 1653	. . . . . . . . 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 555	. . . . . . . 8 |- (Lim B -> (E.x e. B P~A e. (R1` suc x) -> E.y e. B P~A e. (R1` y)))
25		fveq2 3730	. . . . . . . . . 10 |- (x = y -> (R1` x) = (R1` y))
26	25	eleq2d 1544	. . . . . . . . 9 |- (x = y -> (P~A e. (R1` x) <-> P~A e. (R1` y)))
27	26	cbvrexv 1804	. . . . . . . 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 390	. . . . . 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 3053	. . . . . . . . . . . 12 |- x (_ suc x
32		r1ord3 4667	. . . . . . . . . . . 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 3074	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
35	33, 34	sylan2b 454	. . . . . . . . . 10 |- ((x e. On /\ x e. On) -> (R1` x) (_ (R1` suc x))
36	35	anidms 436	. . . . . . . . 9 |- (x e. On -> (R1` x) (_ (R1` suc x))
37	36	sseld 2070	. . . . . . . 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 1740	. . . . . 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 518	. . . . 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 546	. . . 4 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
42	1	eleq2d 1544	. . . . 5 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> P~A e. U_x e. B (R1` x)))
43		eliun 2574	. . . . 5 |- (P~A e. U_x e. B (R1` x) <-> E.x e. B P~A e. (R1` x))
44	42, 43	syl6bb 538	. . . 4 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
45	41, 44	bitr4d 533	. . 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 2288	. . . . 5 |- (A e. (R1` B) -> -. (R1` B) = (/))
48		fvprc 3727	. . . . 5 |- (-. B e. V -> (R1` B) = (/))
49	47, 48	nsyl2 118	. . . 4 |- (A e. (R1` B) -> B e. V)
50		n0i 2288	. . . . 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 680	. . 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 958   e. wcel 960  E.wex 982  E.wrex 1649  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  U_ciun 2570  Oncon0 2954  Lim wlim 2955  suc csuc 2956  ` cfv 3188  R1cr1 4651

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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