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Theorem r1pwcl 5798
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) <-> ~PA e. (R1` B)))
Proof of Theorem r1pwcl
StepHypRef Expression
1 r1lim 5764 . . . . . . 7 |- ((B e. _V /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
21eleq2d 1964 . . . . . 6 |- ((B e. _V /\ Lim B) -> (A e. (R1` B) <-> A e. U_x e. B (R1` x)))
3 eliun 3259 . . . . . 6 |- (A e. U_x e. B (R1` x) <-> E.x e. B A e. (R1` x))
42, 3syl6bb 595 . . . . 5 |- ((B e. _V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
5 onelon 3683 . . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6 limelon 3727 . . . . . . . 8 |- ((B e. _V /\ Lim B) -> B e. On)
75, 6sylan 497 . . . . . . 7 |- (((B e. _V /\ Lim B) /\ x e. B) -> x e. On)
8 r1pw 5797 . . . . . . 7 |- (x e. On -> (A e. (R1` x) <-> ~PA e. (R1` suc x)))
97, 8syl 12 . . . . . 6 |- (((B e. _V /\ Lim B) /\ x e. B) -> (A e. (R1` x) <-> ~PA e. (R1` suc x)))
109rexbidva 2120 . . . . 5 |- ((B e. _V /\ Lim B) -> (E.x e. B A e. (R1` x) <-> E.x e. B ~PA e. (R1` suc x)))
11 limsuc 3933 . . . . . . . . . . . 12 |- (Lim B -> (x e. B <-> suc x e. B))
1211anbi1d 679 . . . . . . . . . . 11 |- (Lim B -> ((x e. B /\ ~PA e. (R1` suc x)) <-> (suc x e. B /\ ~PA e. (R1` suc x))))
13 visset 2295 . . . . . . . . . . . . 13 |- x e. _V
1413sucex 3892 . . . . . . . . . . . 12 |- suc x e. _V
15 eleq1 1957 . . . . . . . . . . . . 13 |- (y = suc x -> (y e. B <-> suc x e. B))
16 fveq2 4681 . . . . . . . . . . . . . 14 |- (y = suc x -> (R1` y) = (R1` suc x))
1716eleq2d 1964 . . . . . . . . . . . . 13 |- (y = suc x -> (~PA e. (R1` y) <-> ~PA e. (R1` suc x)))
1815, 17anbi12d 690 . . . . . . . . . . . 12 |- (y = suc x -> ((y e. B /\ ~PA e. (R1` y)) <-> (suc x e. B /\ ~PA e. (R1` suc x))))
1914, 18cla4ev 2371 . . . . . . . . . . 11 |- ((suc x e. B /\ ~PA e. (R1` suc x)) -> E.y(y e. B /\ ~PA e. (R1` y)))
2012, 19syl6bi 231 . . . . . . . . . 10 |- (Lim B -> ((x e. B /\ ~PA e. (R1` suc x)) -> E.y(y e. B /\ ~PA e. (R1` y))))
212019.23adv 1584 . . . . . . . . 9 |- (Lim B -> (E.x(x e. B /\ ~PA e. (R1` suc x)) -> E.y(y e. B /\ ~PA e. (R1` y))))
22 df-rex 2110 . . . . . . . . 9 |- (E.x e. B ~PA e. (R1` suc x) <-> E.x(x e. B /\ ~PA e. (R1` suc x)))
23 df-rex 2110 . . . . . . . . 9 |- (E.y e. B ~PA e. (R1` y) <-> E.y(y e. B /\ ~PA e. (R1` y)))
2421, 22, 233imtr4g 612 . . . . . . . 8 |- (Lim B -> (E.x e. B ~PA e. (R1` suc x) -> E.y e. B ~PA e. (R1` y)))
25 fveq2 4681 . . . . . . . . . 10 |- (x = y -> (R1` x) = (R1` y))
2625eleq2d 1964 . . . . . . . . 9 |- (x = y -> (~PA e. (R1` x) <-> ~PA e. (R1` y)))
2726cbvrexv 2281 . . . . . . . 8 |- (E.x e. B ~PA e. (R1` x) <-> E.y e. B ~PA e. (R1` y))
2824, 27syl6ibr 230 . . . . . . 7 |- (Lim B -> (E.x e. B ~PA e. (R1` suc x) -> E.x e. B ~PA e. (R1` x)))
2928adantl 424 . . . . . 6 |- ((B e. _V /\ Lim B) -> (E.x e. B ~PA e. (R1` suc x) -> E.x e. B ~PA e. (R1` x)))
307ex 402 . . . . . . . 8 |- ((B e. _V /\ Lim B) -> (x e. B -> x e. On))
31 sssucid 3742 . . . . . . . . . . . 12 |- x C_ suc x
32 r1ord3 5768 . . . . . . . . . . . 12 |- ((x e. On /\ suc x e. On) -> (x C_ suc x -> (R1` x) C_ (R1` suc x)))
3331, 32mpi 55 . . . . . . . . . . 11 |- ((x e. On /\ suc x e. On) -> (R1` x) C_ (R1` suc x))
34 sucelon 3898 . . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
3533, 34sylan2b 501 . . . . . . . . . 10 |- ((x e. On /\ x e. On) -> (R1` x) C_ (R1` suc x))
3635anidms 480 . . . . . . . . 9 |- (x e. On -> (R1` x) C_ (R1` suc x))
3736sseld 2619 . . . . . . . 8 |- (x e. On -> (~PA e. (R1` x) -> ~PA e. (R1` suc x)))
3830, 37syl6 25 . . . . . . 7 |- ((B e. _V /\ Lim B) -> (x e. B -> (~PA e. (R1` x) -> ~PA e. (R1` suc x))))
3938reximdvai 2201 . . . . . 6 |- ((B e. _V /\ Lim B) -> (E.x e. B ~PA e. (R1` x) -> E.x e. B ~PA e. (R1` suc x)))
4029, 39impbid 574 . . . . 5 |- ((B e. _V /\ Lim B) -> (E.x e. B ~PA e. (R1` suc x) <-> E.x e. B ~PA e. (R1` x)))
414, 10, 403bitrd 603 . . . 4 |- ((B e. _V /\ Lim B) -> (A e. (R1` B) <-> E.x e. B ~PA e. (R1` x)))
421eleq2d 1964 . . . . 5 |- ((B e. _V /\ Lim B) -> (~PA e. (R1` B) <-> ~PA e. U_x e. B (R1` x)))
43 eliun 3259 . . . . 5 |- (~PA e. U_x e. B (R1` x) <-> E.x e. B ~PA e. (R1` x))
4442, 43syl6bb 595 . . . 4 |- ((B e. _V /\ Lim B) -> (~PA e. (R1` B) <-> E.x e. B ~PA e. (R1` x)))
4541, 44bitr4d 590 . . 3 |- ((B e. _V /\ Lim B) -> (A e. (R1` B) <-> ~PA e. (R1` B)))
4645ex 402 . 2 |- (B e. _V -> (Lim B -> (A e. (R1` B) <-> ~PA e. (R1` B))))
47 n0i 2880 . . . . 5 |- (A e. (R1` B) -> -. (R1` B) = (/))
48 fvprc 4678 . . . . 5 |- (-. B e. _V -> (R1` B) = (/))
4947, 48nsyl2 133 . . . 4 |- (A e. (R1` B) -> B e. _V)
50 n0i 2880 . . . . 5 |- (~PA e. (R1` B) -> -. (R1` B) = (/))
5150, 48nsyl2 133 . . . 4 |- (~PA e. (R1` B) -> B e. _V)
5249, 51pm5.21ni 742 . . 3 |- (-. B e. _V -> (A e. (R1` B) <-> ~PA e. (R1` B)))
5352a1d 15 . 2 |- (-. B e. _V -> (Lim B -> (A e. (R1` B) <-> ~PA e. (R1` B))))
5446, 53pm2.61i 140 1 |- (Lim B -> (A e. (R1` B) <-> ~PA e. (R1` B)))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U_ciun 3255  Oncon0 3657  Lim wlim 3658  suc csuc 3659  ` cfv 3998  R1cr1 5748
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751
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