Theorem r1pw 8266

Description: A stronger property of R1 than rankpw 8264. 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.) (Revised by Mario Carneiro, 17-Nov-2014.)

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		rankpwi 8244	. . . . . 6 |- ( A e. U. ( R1 " On ) -> ( rank ` ~P A ) = suc ( rank ` A ) )
2	1	eleq1d 2512	. . . . 5 |- ( A e. U. ( R1 " On ) -> ( ( rank ` ~P A ) e. suc B <-> suc ( rank ` A ) e. suc B ) )
3		eloni 4878	. . . . . . 7 |- ( B e. On -> Ord B )
4		ordsucelsuc 6642	. . . . . . 7 |- ( Ord B -> ( ( rank ` A ) e. B <-> suc ( rank ` A ) e. suc B ) )
5	3, 4	syl 16	. . . . . 6 |- ( B e. On -> ( ( rank ` A ) e. B <-> suc ( rank ` A ) e. suc B ) )
6	5	bicomd 201	. . . . 5 |- ( B e. On -> ( suc ( rank ` A ) e. suc B <-> ( rank ` A ) e. B ) )
7	2, 6	sylan9bb 699	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( ( rank ` ~P A ) e. suc B <-> ( rank ` A ) e. B ) )
8		pwwf 8228	. . . . . 6 |- ( A e. U. ( R1 " On ) <-> ~P A e. U. ( R1 " On ) )
9	8	biimpi 194	. . . . 5 |- ( A e. U. ( R1 " On ) -> ~P A e. U. ( R1 " On ) )
10		suceloni 6633	. . . . . 6 |- ( B e. On -> suc B e. On )
11		r1fnon 8188	. . . . . . 7 |- R1 Fn On
12		fndm 5670	. . . . . . 7 |- ( R1 Fn On -> dom R1 = On )
13	11, 12	ax-mp 5	. . . . . 6 |- dom R1 = On
14	10, 13	syl6eleqr 2542	. . . . 5 |- ( B e. On -> suc B e. dom R1 )
15		rankr1ag 8223	. . . . 5 |- ( ( ~P A e. U. ( R1 " On ) /\ suc B e. dom R1 ) -> ( ~P A e. ( R1 ` suc B ) <-> ( rank ` ~P A ) e. suc B ) )
16	9, 14, 15	syl2an 477	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( ~P A e. ( R1 ` suc B ) <-> ( rank ` ~P A ) e. suc B ) )
17	13	eleq2i 2521	. . . . 5 |- ( B e. dom R1 <-> B e. On )
18		rankr1ag 8223	. . . . 5 |- ( ( A e. U. ( R1 " On ) /\ B e. dom R1 ) -> ( A e. ( R1 ` B ) <-> ( rank ` A ) e. B ) )
19	17, 18	sylan2br 476	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( A e. ( R1 ` B ) <-> ( rank ` A ) e. B ) )
20	7, 16, 19	3bitr4rd 286	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )
21	20	ex 434	. 2 |- ( A e. U. ( R1 " On ) -> ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) ) )
22		r1elwf 8217	. . . 4 |- ( A e. ( R1 ` B ) -> A e. U. ( R1 " On ) )
23		r1elwf 8217	. . . . . 6 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. U. ( R1 " On ) )
24		r1elssi 8226	. . . . . 6 |- ( ~P A e. U. ( R1 " On ) -> ~P A C_ U. ( R1 " On ) )
25	23, 24	syl 16	. . . . 5 |- ( ~P A e. ( R1 ` suc B ) -> ~P A C_ U. ( R1 " On ) )
26		ssid 3508	. . . . . 6 |- A C_ A
27		elex 3104	. . . . . . . 8 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. _V )
28		pwexb 6596	. . . . . . . 8 |- ( A e. _V <-> ~P A e. _V )
29	27, 28	sylibr 212	. . . . . . 7 |- ( ~P A e. ( R1 ` suc B ) -> A e. _V )
30		elpwg 4005	. . . . . . 7 |- ( A e. _V -> ( A e. ~P A <-> A C_ A ) )
31	29, 30	syl 16	. . . . . 6 |- ( ~P A e. ( R1 ` suc B ) -> ( A e. ~P A <-> A C_ A ) )
32	26, 31	mpbiri 233	. . . . 5 |- ( ~P A e. ( R1 ` suc B ) -> A e. ~P A )
33	25, 32	sseldd 3490	. . . 4 |- ( ~P A e. ( R1 ` suc B ) -> A e. U. ( R1 " On ) )
34	22, 33	pm5.21ni 352	. . 3 |- ( -. A e. U. ( R1 " On ) -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )
35	34	a1d 25	. 2 |- ( -. A e. U. ( R1 " On ) -> ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) ) )
36	21, 35	pm2.61i 164	1 |- ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   _Vcvv 3095   C_ wss 3461   ~Pcpw 3997   U.cuni 4234   Ord word 4867   Oncon0 4868   suc csuc 4870   dom cdm 4989   "cima 4992   Fn wfn 5573   ` cfv 5578   R1cr1 8183   rankcrnk 8184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186

This theorem is referenced by:  inatsk  9159