Theorem elhf2 29994

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 29993	. 2 |- ( A e. Hf <-> E. x e. om A e. ( R1 ` x ) )
2		omon 6710	. . 3 |- ( om e. On \/ om = On )
3		nnon 6705	. . . . . . . . 9 |- ( x e. om -> x e. On )
4		elhf2.1	. . . . . . . . . 10 |- A e. _V
5	4	rankr1a 8271	. . . . . . . . 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 6709	. . . . . . . . 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 2949	. . . . 5 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
13		peano2 6719	. . . . . . . 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 8294	. . . . . . . . . 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 4021	. . . . . . . . 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 6705	. . . . . . . . . 10 |- ( ( rank ` A ) e. om -> ( rank ` A ) e. On )
20		r1suc 8205	. . . . . . . . . 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 2548	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
24		fveq2 5872	. . . . . . . . 9 |- ( x = suc ( rank ` A ) -> ( R1 ` x ) = ( R1 ` suc ( rank ` A ) ) )
25	24	eleq2d 2527	. . . . . . . 8 |- ( x = suc ( rank ` A ) -> ( A e. ( R1 ` x ) <-> A e. ( R1 ` suc ( rank ` A ) ) ) )
26	25	rspcev 3210	. . . . . . 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 8226	. . . . . 6 |- E. x e. On A e. ( R1 ` x )
31		rankon 8230	. . . . . 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 3055	. . . . . 6 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> E. x e. On A e. ( R1 ` x ) ) )
34		eleq2 2530	. . . . . 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 1395   e. wcel 1819   E.wrex 2808   _Vcvv 3109   C_ wss 3471   ~Pcpw 4015   Oncon0 4887   suc csuc 4889   ` cfv 5594   omcom 6699   R1cr1 8197   rankcrnk 8198   Hf chf 29991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-r1 8199  df-rank 8200  df-hf 29992

This theorem is referenced by:  elhf2g  29995  hfsn  29998