Theorem r1pw 4686

Description: A stronger property of R1 than rankpw 4684. 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 1534	. . . . 5 |- (x = A -> (x e. (R1` B) <-> A e. (R1` B)))
2		pweq 2403	. . . . . 6 |- (x = A -> P~x = P~A)
3	2	eleq1d 1540	. . . . 5 |- (x = A -> (P~x e. (R1` suc B) <-> P~A e. (R1` suc B)))
4	1, 3	bibi12d 629	. . . 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 612	. . 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 1813	. . . . . . 7 |- x e. V
7	6	rankr1a 4677	. . . . . 6 |- (B e. On -> (x e. (R1` B) <-> (rank` x) e. B))
8		eloni 2958	. . . . . . 7 |- (B e. On -> Ord B)
9		ordsucelsuc 3073	. . . . . . 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 528	. . . . 5 |- (B e. On -> (x e. (R1` B) <-> suc (rank` x) e. suc B))
12	6	rankpw 4684	. . . . . 6 |- (rank` P~x) = suc (rank` x)
13	12	eleq1i 1537	. . . . 5 |- ((rank` P~x) e. suc B <-> suc (rank` x) e. suc B)
14	11, 13	syl6bbr 538	. . . 4 |- (B e. On -> (x e. (R1` B) <-> (rank` P~x) e. suc B))
15		suceloni 3062	. . . . 5 |- (B e. On -> suc B e. On)
16	6	pwex 2745	. . . . . 6 |- P~x e. V
17	16	rankr1a 4677	. . . . 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 531	. . 3 |- (B e. On -> (x e. (R1` B) <-> P~x e. (R1` suc B)))
20	5, 19	vtoclg 1847	. 2 |- (A e. V -> (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B))))
21		elisset 1817	. . . 4 |- (A e. (R1` B) -> A e. V)
22		elisset 1817	. . . . 5 |- (P~A e. (R1` suc B) -> P~A e. V)
23		pwexb 2908	. . . . 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 678	. . 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 956   e. wcel 958  Vcvv 1811  P~cpw 2401  Ord word 2947  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  r1pwcl 4687

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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