Theorem r1pw 7727

Description: A stronger property of R1 than rankpw 7725. 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 7705	. . . . . 6 |- ( A e. U. ( R1 " On ) -> ( rank ` ~P A ) = suc ( rank ` A ) )
2	1	eleq1d 2470	. . . . 5 |- ( A e. U. ( R1 " On ) -> ( ( rank ` ~P A ) e. suc B <-> suc ( rank ` A ) e. suc B ) )
3		eloni 4551	. . . . . . 7 |- ( B e. On -> Ord B )
4		ordsucelsuc 4761	. . . . . . 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 193	. . . . 5 |- ( B e. On -> ( suc ( rank ` A ) e. suc B <-> ( rank ` A ) e. B ) )
7	2, 6	sylan9bb 681	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( ( rank ` ~P A ) e. suc B <-> ( rank ` A ) e. B ) )
8		pwwf 7689	. . . . . 6 |- ( A e. U. ( R1 " On ) <-> ~P A e. U. ( R1 " On ) )
9	8	biimpi 187	. . . . 5 |- ( A e. U. ( R1 " On ) -> ~P A e. U. ( R1 " On ) )
10		suceloni 4752	. . . . . 6 |- ( B e. On -> suc B e. On )
11		r1fnon 7649	. . . . . . 7 |- R1 Fn On
12		fndm 5503	. . . . . . 7 |- ( R1 Fn On -> dom R1 = On )
13	11, 12	ax-mp 8	. . . . . 6 |- dom R1 = On
14	10, 13	syl6eleqr 2495	. . . . 5 |- ( B e. On -> suc B e. dom R1 )
15		rankr1ag 7684	. . . . 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 464	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( ~P A e. ( R1 ` suc B ) <-> ( rank ` ~P A ) e. suc B ) )
17	13	eleq2i 2468	. . . . 5 |- ( B e. dom R1 <-> B e. On )
18		rankr1ag 7684	. . . . 5 |- ( ( A e. U. ( R1 " On ) /\ B e. dom R1 ) -> ( A e. ( R1 ` B ) <-> ( rank ` A ) e. B ) )
19	17, 18	sylan2br 463	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( A e. ( R1 ` B ) <-> ( rank ` A ) e. B ) )
20	7, 16, 19	3bitr4rd 278	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. On ) -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )
21	20	ex 424	. 2 |- ( A e. U. ( R1 " On ) -> ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) ) )
22		r1elwf 7678	. . . 4 |- ( A e. ( R1 ` B ) -> A e. U. ( R1 " On ) )
23		r1elwf 7678	. . . . . 6 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. U. ( R1 " On ) )
24		r1elssi 7687	. . . . . 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 3327	. . . . . 6 |- A C_ A
27		elex 2924	. . . . . . . 8 |- ( ~P A e. ( R1 ` suc B ) -> ~P A e. _V )
28		pwexb 4712	. . . . . . . 8 |- ( A e. _V <-> ~P A e. _V )
29	27, 28	sylibr 204	. . . . . . 7 |- ( ~P A e. ( R1 ` suc B ) -> A e. _V )
30		elpwg 3766	. . . . . . 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 225	. . . . 5 |- ( ~P A e. ( R1 ` suc B ) -> A e. ~P A )
33	25, 32	sseldd 3309	. . . 4 |- ( ~P A e. ( R1 ` suc B ) -> A e. U. ( R1 " On ) )
34	22, 33	pm5.21ni 342	. . 3 |- ( -. A e. U. ( R1 " On ) -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )
35	34	a1d 23	. 2 |- ( -. A e. U. ( R1 " On ) -> ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) ) )
36	21, 35	pm2.61i 158	1 |- ( B e. On -> ( A e. ( R1 ` B ) <-> ~P A e. ( R1 ` suc B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   C_ wss 3280   ~Pcpw 3759   U.cuni 3975   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837   "cima 4840   Fn wfn 5408   ` cfv 5413   R1cr1 7644   rankcrnk 7645

This theorem is referenced by:  inatsk  8609

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647