Theorem r1pwcl 3530

Description: The cumulative hierarchy of a limit ordinal is closed under powerset. (Contributed by Raph Levien, 29-May-04.)

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 3497	. . . . . . 7 |- ((B e. V /\ Lim B) -> (R1` B) = U.x e. B (R1` x))
2	1	eleq2d 1156	. . . . . 6 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> A e. U.x e. B (R1` x)))
3		eliun 1998	. . . . . 6 |- (A e. U.x e. B (R1` x) <-> E.x e. B A e. (R1` x))
4	2, 3	syl6bb 414	. . . . 5 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
5		onelon 2223	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6		limelon 2286	. . . . . . . 8 |- ((B e. V /\ Lim B) -> B e. On)
7	5, 6	sylan 343	. . . . . . 7 |- (((B e. V /\ Lim B) /\ x e. B) -> x e. On)
8		r1pw 3529	. . . . . . 7 |- (x e. On -> (A e. (R1` x) <-> P~A e. (R1` suc x)))
9	7, 8	syl 12	. . . . . 6 |- (((B e. V /\ Lim B) /\ x e. B) -> (A e. (R1` x) <-> P~A e. (R1` suc x)))
10	9	birexdva 1216	. . . . 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 2361	. . . . . . . . . . . 12 |- (Lim B -> (x e. B <-> suc x e. B))
12	11	anbi1d 469	. . . . . . . . . . 11 |- (Lim B -> ((x e. B /\ P~A e. (R1` suc x)) <-> (suc x e. B /\ P~A e. (R1` suc x))))
13		visset 1350	. . . . . . . . . . . . 13 |- x e. V
14	13	sucex 2303	. . . . . . . . . . . 12 |- suc x e. V
15		eleq1 1149	. . . . . . . . . . . . 13 |- (y = suc x -> (y e. B <-> suc x e. B))
16		fveq2 2832	. . . . . . . . . . . . . 14 |- (y = suc x -> (R1` y) = (R1` suc x))
17	16	eleq2d 1156	. . . . . . . . . . . . 13 |- (y = suc x -> (P~A e. (R1` y) <-> P~A e. (R1` suc x)))
18	15, 17	anbi12d 476	. . . . . . . . . . . 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 1401	. . . . . . . . . . 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 187	. . . . . . . . . 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 954	. . . . . . . . 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 1206	. . . . . . . . 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 1206	. . . . . . . . 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 426	. . . . . . . 8 |- (Lim B -> (E.x e. B P~A e. (R1` suc x) -> E.y e. B P~A e. (R1` y)))
25		fveq2 2832	. . . . . . . . . 10 |- (x = y -> (R1` x) = (R1` y))
26	25	eleq2d 1156	. . . . . . . . 9 |- (x = y -> (P~A e. (R1` x) <-> P~A e. (R1` y)))
27	26	cbvrexv 1334	. . . . . . . 8 |- (E.x e. B P~A e. (R1` x) <-> E.y e. B P~A e. (R1` y))
28	24, 27	syl6ibr 186	. . . . . . 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 305	. . . . . 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	exp 291	. . . . . . . 8 |- ((B e. V /\ Lim B) -> (x e. B -> x e. On))
31		sssucid 2300	. . . . . . . . . . . 12 |- x (_ suc x
32		r1ord3 3501	. . . . . . . . . . . 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 2319	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
35	33, 34	sylan2b 347	. . . . . . . . . 10 |- ((x e. On /\ x e. On) -> (R1` x) (_ (R1` suc x))
36	35	anidms 332	. . . . . . . . 9 |- (x e. On -> (R1` x) (_ (R1` suc x))
37	36	sseld 1506	. . . . . . . 8 |- (x e. On -> (P~A e. (R1` x) -> P~A e. (R1` suc x)))
38	30, 37	syl6 23	. . . . . . 7 |- ((B e. V /\ Lim B) -> (x e. B -> (P~A e. (R1` x) -> P~A e. (R1` suc x))))
39	38	r19.22dv 1278	. . . . . 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 397	. . . . 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 422	. . . 4 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
42	1	eleq2d 1156	. . . . 5 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> P~A e. U.x e. B (R1` x)))
43		eliun 1998	. . . . 5 |- (P~A e. U.x e. B (R1` x) <-> E.x e. B P~A e. (R1` x))
44	42, 43	syl6bb 414	. . . 4 |- ((B e. V /\ Lim B) -> (P~A e. (R1` B) <-> E.x e. B P~A e. (R1` x)))
45	41, 44	bitr4d 409	. . 3 |- ((B e. V /\ Lim B) -> (A e. (R1` B) <-> P~A e. (R1` B)))
46	45	exp 291	. 2 |- (B e. V -> (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B))))
47		n0i 1712	. . . . 5 |- (A e. (R1` B) -> -. (R1` B) = (/))
48		fvprc 2829	. . . . 5 |- (-. B e. V -> (R1` B) = (/))
49	47, 48	nsyl2 103	. . . 4 |- (A e. (R1` B) -> B e. V)
50		n0i 1712	. . . . 5 |- (P~A e. (R1` B) -> -. (R1` B) = (/))
51	50, 48	nsyl2 103	. . . 4 |- (P~A e. (R1` B) -> B e. V)
52	49, 51	pm5.21ni 503	. . 3 |- (-. B e. V -> (A e. (R1` B) <-> P~A e. (R1` B)))
53	52	a1d 14	. 2 |- (-. B e. V -> (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B))))
54	46, 53	pm2.61i 110	1 |- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348   (_ wss 1487  (/)c0 1707  P~cpw 1798  U.ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  ` cfv 2422  R1cr1 3485

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

metamath.org