Theorem r1pwcl 4687

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 4653	. . . . . . 7 |- ((B e. V /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
2	1	eleq2d 1541	. . . . . 6 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> A e. U_x e. B (R1` x)))
3		eliun 2570	. . . . . 6 |- (A e. U_x e. B (R1` x) <-> E.x e. B A e. (R1` x))
4	2, 3	syl6bb 536	. . . . 5 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
5		onelon 2972	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6		limelon 3032	. . . . . . . 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 4686	. . . . . . 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 1660	. . . . 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 3120	. . . . . . . . . . . 12 |- (Lim B -> (x e. B <-> suc x e. B))
12	11	anbi1d 617	. . . . . . . . . . 11 |- (Lim B -> ((x e. B /\ P~A e. (R1` suc x)) <-> (suc x e. B /\ P~A e. (R1` suc x))))
13		visset 1813	. . . . . . . . . . . . 13 |- x e. V
14	13	sucex 3050	. . . . . . . . . . . 12 |- suc x e. V
15		eleq1 1534	. . . . . . . . . . . . 13 |- (y = suc x -> (y e. B <-> suc x e. B))
16		fveq2 3724	. . . . . . . . . . . . . 14 |- (y = suc x -> (R1` y) = (R1` suc x))
17	16	eleq2d 1541	. . . . . . . . . . . . 13 |- (y = suc x -> (P~A e. (R1` y) <-> P~A e. (R1` suc x)))
18	15, 17	anbi12d 628	. . . . . . . . . . . 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 1869	. . . . . . . . . . 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 1214	. . . . . . . . 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 1650	. . . . . . . . 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 1650	. . . . . . . . 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 553	. . . . . . . 8 |- (Lim B -> (E.x e. B P~A e. (R1` suc x) -> E.y e. B P~A e. (R1` y)))
25		fveq2 3724	. . . . . . . . . 10 |- (x = y -> (R1` x) = (R1` y))
26	25	eleq2d 1541	. . . . . . . . 9 |- (x = y -> (P~A e. (R1` x) <-> P~A e. (R1` y)))
27	26	cbvrexv 1801	. . . . . . . 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 3047	. . . . . . . . . . . 12 |- x (_ suc x
32		r1ord3 4657	. . . . . . . . . . . 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 3068	. . . . . . . . . . 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 2067	. . . . . . . 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 1737	. . . . . 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 516	. . . . 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 544	. . . 4 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
42	1	eleq2d 1541	. . . . 5 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> P~A e. U_x e. B (R1` x)))
43		eliun 2570	. . . . 5 |- (P~A e. U_x e. B (R1` x) <-> E.x e. B P~A e. (R1` x))
44	42, 43	syl6bb 536	. . . 4 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
45	41, 44	bitr4d 531	. . 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 2285	. . . . 5 |- (A e. (R1` B) -> -. (R1` B) = (/))
48		fvprc 3721	. . . . 5 |- (-. B e. V -> (R1` B) = (/))
49	47, 48	nsyl2 118	. . . 4 |- (A e. (R1` B) -> B e. V)
50		n0i 2285	. . . . 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 678	. . 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 956   e. wcel 958  E.wex 980  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  U_ciun 2566  Oncon0 2948  Lim wlim 2949  suc csuc 2950  ` cfv 3182  R1cr1 4641

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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