Theorem elhf2 32282

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 32281	. 2 |- ( A e. Hf <-> E. x e. om A e. ( R1 ` x ) )
2		omon 7076	. . 3 |- ( om e. On \/ om = On )
3		nnon 7071	. . . . . . . . 9 |- ( x e. om -> x e. On )
4		elhf2.1	. . . . . . . . . 10 |- A e. _V
5	4	rankr1a 8699	. . . . . . . . 9 |- ( x e. On -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
6	3, 5	syl 17	. . . . . . . 8 |- ( x e. om -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
7	6	adantl 482	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
8		elnn 7075	. . . . . . . . 9 |- ( ( ( rank ` A ) e. x /\ x e. om ) -> ( rank ` A ) e. om )
9	8	expcom 451	. . . . . . . 8 |- ( x e. om -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
10	9	adantl 482	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
11	7, 10	sylbid 230	. . . . . 6 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
12	11	rexlimdva 3031	. . . . 5 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
13		peano2 7086	. . . . . . . 8 |- ( ( rank ` A ) e. om -> suc ( rank ` A ) e. om )
14	13	adantr 481	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> suc ( rank ` A ) e. om )
15		r1rankid 8722	. . . . . . . . . 10 |- ( A e. _V -> A C_ ( R1 ` ( rank ` A ) ) )
16	4, 15	mp1i 13	. . . . . . . . 9 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A C_ ( R1 ` ( rank ` A ) ) )
17	4	elpw 4164	. . . . . . . . 9 |- ( A e. ~P ( R1 ` ( rank ` A ) ) <-> A C_ ( R1 ` ( rank ` A ) ) )
18	16, 17	sylibr 224	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ~P ( R1 ` ( rank ` A ) ) )
19		nnon 7071	. . . . . . . . . 10 |- ( ( rank ` A ) e. om -> ( rank ` A ) e. On )
20		r1suc 8633	. . . . . . . . . 10 |- ( ( rank ` A ) e. On -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
21	19, 20	syl 17	. . . . . . . . 9 |- ( ( rank ` A ) e. om -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
22	21	adantr 481	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
23	18, 22	eleqtrrd 2704	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
24		fveq2 6191	. . . . . . . . 9 |- ( x = suc ( rank ` A ) -> ( R1 ` x ) = ( R1 ` suc ( rank ` A ) ) )
25	24	eleq2d 2687	. . . . . . . 8 |- ( x = suc ( rank ` A ) -> ( A e. ( R1 ` x ) <-> A e. ( R1 ` suc ( rank ` A ) ) ) )
26	25	rspcev 3309	. . . . . . 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 693	. . . . . 6 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> E. x e. om A e. ( R1 ` x ) )
28	27	expcom 451	. . . . 5 |- ( om e. On -> ( ( rank ` A ) e. om -> E. x e. om A e. ( R1 ` x ) ) )
29	12, 28	impbid 202	. . . 4 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
30	4	tz9.13 8654	. . . . . 6 |- E. x e. On A e. ( R1 ` x )
31		rankon 8658	. . . . . 6 |- ( rank ` A ) e. On
32	30, 31	2th 254	. . . . 5 |- ( E. x e. On A e. ( R1 ` x ) <-> ( rank ` A ) e. On )
33		rexeq 3139	. . . . . 6 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> E. x e. On A e. ( R1 ` x ) ) )
34		eleq2 2690	. . . . . 6 |- ( om = On -> ( ( rank ` A ) e. om <-> ( rank ` A ) e. On ) )
35	33, 34	bibi12d 335	. . . . 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 248	. . . 4 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
37	29, 36	jaoi 394	. . 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 264	1 |- ( A e. Hf <-> ( rank ` A ) e. om )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   Oncon0 5723   suc csuc 5725   ` cfv 5888   omcom 7065   R1cr1 8625   rankcrnk 8626   Hf chf 32279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-r1 8627  df-rank 8628  df-hf 32280

This theorem is referenced by:  elhf2g  32283  hfsn  32286