Theorem r1pw 4666

Description: A stronger property of R1 than rankpw 4664. 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 P~A 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) <-> P~A e. (R1` suc B)))

Proof of Theorem r1pw

Step	Hyp	Ref	Expression
1		eleq1 1531	. . . . 5 |- (x = A -> (x e. (R1` B) <-> A e. (R1` B)))
2		pweq 2399	. . . . . 6 |- (x = A -> P~x = P~A)
3	2	eleq1d 1537	. . . . 5 |- (x = A -> (P~x e. (R1` suc B) <-> P~A e. (R1` suc B)))
4	1, 3	bibi12d 628	. . . 4 |- (x = A -> ((x e. (R1` B) <-> P~x e. (R1` suc B)) <-> (A e. (R1` B) <-> P~A e. (R1` suc B))))
5	4	imbi2d 611	. . 3 |- (x = A -> ((B e. On -> (x e. (R1` B) <-> P~x e. (R1` suc B))) <-> (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B)))))
6		visset 1809	. . . . . . 7 |- x e. V
7	6	rankr1a 4657	. . . . . 6 |- (B e. On -> (x e. (R1` B) <-> (rank` x) e. B))
8		eloni 2953	. . . . . . 7 |- (B e. On -> Ord B)
9		ordsucelsuc 3068	. . . . . . 7 |- (Ord B -> ((rank` x) e. B <-> suc (rank` x) e. suc B))
10	8, 9	syl 10	. . . . . 6 |- (B e. On -> ((rank` x) e. B <-> suc (rank` x) e. suc B))
11	7, 10	bitrd 527	. . . . 5 |- (B e. On -> (x e. (R1` B) <-> suc (rank` x) e. suc B))
12	6	rankpw 4664	. . . . . 6 |- (rank` P~x) = suc (rank` x)
13	12	eleq1i 1534	. . . . 5 |- ((rank` P~x) e. suc B <-> suc (rank` x) e. suc B)
14	11, 13	syl6bbr 537	. . . 4 |- (B e. On -> (x e. (R1` B) <-> (rank` P~x) e. suc B))
15		suceloni 3057	. . . . 5 |- (B e. On -> suc B e. On)
16	6	pwex 2740	. . . . . 6 |- P~x e. V
17	16	rankr1a 4657	. . . . 5 |- (suc B e. On -> (P~x e. (R1` suc B) <-> (rank` P~x) e. suc B))
18	15, 17	syl 10	. . . 4 |- (B e. On -> (P~x e. (R1` suc B) <-> (rank` P~x) e. suc B))
19	14, 18	bitr4d 530	. . 3 |- (B e. On -> (x e. (R1` B) <-> P~x e. (R1` suc B)))
20	5, 19	vtoclg 1843	. 2 |- (A e. V -> (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B))))
21		elisset 1813	. . . 4 |- (A e. (R1` B) -> A e. V)
22		elisset 1813	. . . . 5 |- (P~A e. (R1` suc B) -> P~A e. V)
23		pwexb 2903	. . . . 5 |- (A e. V <-> P~A e. V)
24	22, 23	sylibr 200	. . . 4 |- (P~A e. (R1` suc B) -> A e. V)
25	21, 24	pm5.21ni 677	. . 3 |- (-. A e. V -> (A e. (R1` B) <-> P~A e. (R1` suc B)))
26	25	a1d 12	. 2 |- (-. A e. V -> (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B))))
27	20, 26	pm2.61i 126	1 |- (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  Vcvv 1807  P~cpw 2397  Ord word 2942  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  r1pwcl 4667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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