Theorem elhf2 28218

Description: Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)

Hypothesis

Ref	Expression
elhf2.1	|- A e. _V

Assertion

Ref	Expression
elhf2	|- ( A e. Hf <-> ( rank ` A ) e. om )

Proof of Theorem elhf2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhf 28217	. 2 |- ( A e. Hf <-> E. x e. om A e. ( R1 ` x ) )
2		omon 6492	. . 3 |- ( om e. On \/ om = On )
3		nnon 6487	. . . . . . . . 9 |- ( x e. om -> x e. On )
4		elhf2.1	. . . . . . . . . 10 |- A e. _V
5	4	rankr1a 8048	. . . . . . . . 9 |- ( x e. On -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
6	3, 5	syl 16	. . . . . . . 8 |- ( x e. om -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
7	6	adantl 466	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
8		elnn 6491	. . . . . . . . 9 |- ( ( ( rank ` A ) e. x /\ x e. om ) -> ( rank ` A ) e. om )
9	8	expcom 435	. . . . . . . 8 |- ( x e. om -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
10	9	adantl 466	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
11	7, 10	sylbid 215	. . . . . 6 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
12	11	rexlimdva 2846	. . . . 5 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
13		peano2 6501	. . . . . . . 8 |- ( ( rank ` A ) e. om -> suc ( rank ` A ) e. om )
14	13	adantr 465	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> suc ( rank ` A ) e. om )
15		r1rankid 8071	. . . . . . . . . 10 |- ( A e. _V -> A C_ ( R1 ` ( rank ` A ) ) )
16	4, 15	mp1i 12	. . . . . . . . 9 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A C_ ( R1 ` ( rank ` A ) ) )
17	4	elpw 3871	. . . . . . . . 9 |- ( A e. ~P ( R1 ` ( rank ` A ) ) <-> A C_ ( R1 ` ( rank ` A ) ) )
18	16, 17	sylibr 212	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ~P ( R1 ` ( rank ` A ) ) )
19		nnon 6487	. . . . . . . . . 10 |- ( ( rank ` A ) e. om -> ( rank ` A ) e. On )
20		r1suc 7982	. . . . . . . . . 10 |- ( ( rank ` A ) e. On -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
21	19, 20	syl 16	. . . . . . . . 9 |- ( ( rank ` A ) e. om -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
22	21	adantr 465	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
23	18, 22	eleqtrrd 2520	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
24		fveq2 5696	. . . . . . . . 9 |- ( x = suc ( rank ` A ) -> ( R1 ` x ) = ( R1 ` suc ( rank ` A ) ) )
25	24	eleq2d 2510	. . . . . . . 8 |- ( x = suc ( rank ` A ) -> ( A e. ( R1 ` x ) <-> A e. ( R1 ` suc ( rank ` A ) ) ) )
26	25	rspcev 3078	. . . . . . 7 |- ( ( suc ( rank ` A ) e. om /\ A e. ( R1 ` suc ( rank ` A ) ) ) -> E. x e. om A e. ( R1 ` x ) )
27	14, 23, 26	syl2anc 661	. . . . . 6 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> E. x e. om A e. ( R1 ` x ) )
28	27	expcom 435	. . . . 5 |- ( om e. On -> ( ( rank ` A ) e. om -> E. x e. om A e. ( R1 ` x ) ) )
29	12, 28	impbid 191	. . . 4 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
30	4	tz9.13 8003	. . . . . 6 |- E. x e. On A e. ( R1 ` x )
31		rankon 8007	. . . . . 6 |- ( rank ` A ) e. On
32	30, 31	2th 239	. . . . 5 |- ( E. x e. On A e. ( R1 ` x ) <-> ( rank ` A ) e. On )
33		rexeq 2923	. . . . . 6 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> E. x e. On A e. ( R1 ` x ) ) )
34		eleq2 2504	. . . . . 6 |- ( om = On -> ( ( rank ` A ) e. om <-> ( rank ` A ) e. On ) )
35	33, 34	bibi12d 321	. . . . 5 |- ( om = On -> ( ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) <-> ( E. x e. On A e. ( R1 ` x ) <-> ( rank ` A ) e. On ) ) )
36	32, 35	mpbiri 233	. . . 4 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
37	29, 36	jaoi 379	. . 3 |- ( ( om e. On \/ om = On ) -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
38	2, 37	ax-mp 5	. 2 |- ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om )
39	1, 38	bitri 249	1 |- ( A e. Hf <-> ( rank ` A ) e. om )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   E.wrex 2721   _Vcvv 2977   C_ wss 3333   ~Pcpw 3865   Oncon0 4724   suc csuc 4726   ` cfv 5423   omcom 6481   R1cr1 7974   rankcrnk 7975   Hf chf 28215

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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852

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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-r1 7976  df-rank 7977  df-hf 28216

This theorem is referenced by:  elhf2g  28219  hfsn  28222