Theorem r1pw 8164

Description: A stronger property of R1 than rankpw 8162. 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 8142	. . . . . 6 |- ( A e. U. ( R1 " On ) -> ( rank ` ~P A ) = suc ( rank ` A ) )
2	1	eleq1d 2523	. . . . 5 |- ( A e. U. ( R1 " On ) -> ( ( rank ` ~P A ) e. suc B <-> suc ( rank ` A ) e. suc B ) )
3		eloni 4838	. . . . . . 7 |- ( B e. On -> Ord B )
4		ordsucelsuc 6544	. . . . . . 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 8126	. . . . . 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 6535	. . . . . 6 |- ( B e. On -> suc B e. On )
11		r1fnon 8086	. . . . . . 7 |- R1 Fn On
12		fndm 5619	. . . . . . 7 |- ( R1 Fn On -> dom R1 = On )
13	11, 12	ax-mp 5	. . . . . 6 |- dom R1 = On
14	10, 13	syl6eleqr 2553	. . . . 5 |- ( B e. On -> suc B e. dom R1 )
15		rankr1ag 8121	. . . . 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 2532	. . . . 5 |- ( B e. dom R1 <-> B e. On )
18		rankr1ag 8121	. . . . 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 8115	. . . 4 |- ( A e. ( R1 ` B ) -> A e. U. ( R1 " On ) )
23		r1elwf 8115	. . . . . 6 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. U. ( R1 " On ) )
24		r1elssi 8124	. . . . . 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 3484	. . . . . 6 |- A C_ A
27		elex 3087	. . . . . . . 8 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. _V )
28		pwexb 6498	. . . . . . . 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 3977	. . . . . . 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 3466	. . . 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 1370   e. wcel 1758   _Vcvv 3078   C_ wss 3437   ~Pcpw 3969   U.cuni 4200   Ord word 4827   Oncon0 4828   suc csuc 4830   dom cdm 4949   "cima 4952   Fn wfn 5522   ` cfv 5527   R1cr1 8081   rankcrnk 8082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-r1 8083  df-rank 8084

This theorem is referenced by:  inatsk  9057