Theorem r1pw 8044

Description: A stronger property of R1 than rankpw 8042. 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 8022	. . . . . 6 |- ( A e. U. ( R1 " On ) -> ( rank ` ~P A ) = suc ( rank ` A ) )
2	1	eleq1d 2504	. . . . 5 |- ( A e. U. ( R1 " On ) -> ( ( rank ` ~P A ) e. suc B <-> suc ( rank ` A ) e. suc B ) )
3		eloni 4724	. . . . . . 7 |- ( B e. On -> Ord B )
4		ordsucelsuc 6428	. . . . . . 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 8006	. . . . . 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 6419	. . . . . 6 |- ( B e. On -> suc B e. On )
11		r1fnon 7966	. . . . . . 7 |- R1 Fn On
12		fndm 5505	. . . . . . 7 |- ( R1 Fn On -> dom R1 = On )
13	11, 12	ax-mp 5	. . . . . 6 |- dom R1 = On
14	10, 13	syl6eleqr 2529	. . . . 5 |- ( B e. On -> suc B e. dom R1 )
15		rankr1ag 8001	. . . . 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 2502	. . . . 5 |- ( B e. dom R1 <-> B e. On )
18		rankr1ag 8001	. . . . 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 7995	. . . 4 |- ( A e. ( R1 ` B ) -> A e. U. ( R1 " On ) )
23		r1elwf 7995	. . . . . 6 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. U. ( R1 " On ) )
24		r1elssi 8004	. . . . . 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 3370	. . . . . 6 |- A C_ A
27		elex 2976	. . . . . . . 8 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. _V )
28		pwexb 6382	. . . . . . . 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 3863	. . . . . . 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 3352	. . . 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 1369   e. wcel 1756   _Vcvv 2967   C_ wss 3323   ~Pcpw 3855   U.cuni 4086   Ord word 4713   Oncon0 4714   suc csuc 4716   dom cdm 4835   "cima 4838   Fn wfn 5408   ` cfv 5413   R1cr1 7961   rankcrnk 7962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964

This theorem is referenced by:  inatsk  8937