Theorem r1pw 5797

Description: A stronger property of R1 than rankpw 5795. The latter merely proves that R1 of the successor is a power set, but here we prove that if A is in the cumulative hierarchy, then ~PA is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.)

Assertion

Ref	Expression
r1pw	|- (B e. On -> (A e. (R1` B) <-> ~PA e. (R1` suc B)))

Proof of Theorem r1pw

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (x = A -> (x e. (R1` B) <-> A e. (R1` B)))
2		pweq 3036	. . . . . 6 |- (x = A -> ~Px = ~PA)
3	2	eleq1d 1963	. . . . 5 |- (x = A -> (~Px e. (R1` suc B) <-> ~PA e. (R1` suc B)))
4	1, 3	bibi12d 691	. . . 4 |- (x = A -> ((x e. (R1` B) <-> ~Px e. (R1` suc B)) <-> (A e. (R1` B) <-> ~PA e. (R1` suc B))))
5	4	imbi2d 674	. . 3 |- (x = A -> ((B e. On -> (x e. (R1` B) <-> ~Px e. (R1` suc B))) <-> (B e. On -> (A e. (R1` B) <-> ~PA e. (R1` suc B)))))
6		visset 2295	. . . . . . 7 |- x e. _V
7	6	rankr1a 5788	. . . . . 6 |- (B e. On -> (x e. (R1` B) <-> (rank` x) e. B))
8		eloni 3667	. . . . . . 7 |- (B e. On -> Ord B)
9		ordsucelsuc 3902	. . . . . . 7 |- (Ord B -> ((rank` x) e. B <-> suc (rank` x) e. suc B))
10	8, 9	syl 12	. . . . . 6 |- (B e. On -> ((rank` x) e. B <-> suc (rank` x) e. suc B))
11	7, 10	bitrd 587	. . . . 5 |- (B e. On -> (x e. (R1` B) <-> suc (rank` x) e. suc B))
12	6	rankpw 5795	. . . . . 6 |- (rank` ~Px) = suc (rank` x)
13	12	eleq1i 1960	. . . . 5 |- ((rank` ~Px) e. suc B <-> suc (rank` x) e. suc B)
14	11, 13	syl6bbr 597	. . . 4 |- (B e. On -> (x e. (R1` B) <-> (rank` ~Px) e. suc B))
15		suceloni 3894	. . . . 5 |- (B e. On -> suc B e. On)
16	6	pwex 3487	. . . . . 6 |- ~Px e. _V
17	16	rankr1a 5788	. . . . 5 |- (suc B e. On -> (~Px e. (R1` suc B) <-> (rank` ~Px) e. suc B))
18	15, 17	syl 12	. . . 4 |- (B e. On -> (~Px e. (R1` suc B) <-> (rank` ~Px) e. suc B))
19	14, 18	bitr4d 590	. . 3 |- (B e. On -> (x e. (R1` B) <-> ~Px e. (R1` suc B)))
20	5, 19	vtoclg 2346	. 2 |- (A e. _V -> (B e. On -> (A e. (R1` B) <-> ~PA e. (R1` suc B))))
21		elisset 2299	. . . 4 |- (A e. (R1` B) -> A e. _V)
22		elisset 2299	. . . . 5 |- (~PA e. (R1` suc B) -> ~PA e. _V)
23		pwexb 3852	. . . . 5 |- (A e. _V <-> ~PA e. _V)
24	22, 23	sylibr 217	. . . 4 |- (~PA e. (R1` suc B) -> A e. _V)
25	21, 24	pm5.21ni 742	. . 3 |- (-. A e. _V -> (A e. (R1` B) <-> ~PA e. (R1` suc B)))
26	25	a1d 15	. 2 |- (-. A e. _V -> (B e. On -> (A e. (R1` B) <-> ~PA e. (R1` suc B))))
27	20, 26	pm2.61i 140	1 |- (B e. On -> (A e. (R1` B) <-> ~PA e. (R1` suc B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292  ~Pcpw 3032  Ord word 3656  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  r1pwcl 5798

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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