Theorem ranklim 8163

Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)

Assertion

Ref	Expression
ranklim	|- ( Lim B -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )

Proof of Theorem ranklim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsuc 6571	. . . 4 |- ( Lim B -> ( ( rank ` A ) e. B <-> suc ( rank ` A ) e. B ) )
2	1	adantl 466	. . 3 |- ( ( A e. _V /\ Lim B ) -> ( ( rank ` A ) e. B <-> suc ( rank ` A ) e. B ) )
3		pweq 3972	. . . . . . . 8 |- ( x = A -> ~P x = ~P A )
4	3	fveq2d 5804	. . . . . . 7 |- ( x = A -> ( rank ` ~P x ) = ( rank ` ~P A ) )
5		fveq2 5800	. . . . . . . 8 |- ( x = A -> ( rank ` x ) = ( rank ` A ) )
6		suceq 4893	. . . . . . . 8 |- ( ( rank ` x ) = ( rank ` A ) -> suc ( rank ` x ) = suc ( rank ` A ) )
7	5, 6	syl 16	. . . . . . 7 |- ( x = A -> suc ( rank ` x ) = suc ( rank ` A ) )
8	4, 7	eqeq12d 2476	. . . . . 6 |- ( x = A -> ( ( rank ` ~P x ) = suc ( rank ` x ) <-> ( rank ` ~P A ) = suc ( rank ` A ) ) )
9		vex 3081	. . . . . . 7 |- x e. _V
10	9	rankpw 8162	. . . . . 6 |- ( rank ` ~P x ) = suc ( rank ` x )
11	8, 10	vtoclg 3136	. . . . 5 |- ( A e. _V -> ( rank ` ~P A ) = suc ( rank ` A ) )
12	11	eleq1d 2523	. . . 4 |- ( A e. _V -> ( ( rank ` ~P A ) e. B <-> suc ( rank ` A ) e. B ) )
13	12	adantr 465	. . 3 |- ( ( A e. _V /\ Lim B ) -> ( ( rank ` ~P A ) e. B <-> suc ( rank ` A ) e. B ) )
14	2, 13	bitr4d 256	. 2 |- ( ( A e. _V /\ Lim B ) -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )
15		fvprc 5794	. . . . 5 |- ( -. A e. _V -> ( rank ` A ) = (/) )
16		pwexb 6498	. . . . . 6 |- ( A e. _V <-> ~P A e. _V )
17		fvprc 5794	. . . . . 6 |- ( -. ~P A e. _V -> ( rank ` ~P A ) = (/) )
18	16, 17	sylnbi 306	. . . . 5 |- ( -. A e. _V -> ( rank ` ~P A ) = (/) )
19	15, 18	eqtr4d 2498	. . . 4 |- ( -. A e. _V -> ( rank ` A ) = ( rank ` ~P A ) )
20	19	eleq1d 2523	. . 3 |- ( -. A e. _V -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )
21	20	adantr 465	. 2 |- ( ( -. A e. _V /\ Lim B ) -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )
22	14, 21	pm2.61ian 788	1 |- ( Lim B -> ( ( rank ` A ) e. B <-> ( rank ` ~P A ) e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3078   (/)c0 3746   ~Pcpw 3969   Lim wlim 4829   suc csuc 4830   ` cfv 5527   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  ax-reg 7919  ax-inf2 7959

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:  rankxplim  8198