Theorem elhf2 30990

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 30989	. 2 |- ( A e. Hf <-> E. x e. om A e. ( R1 ` x ) )
2		omon 6729	. . 3 |- ( om e. On \/ om = On )
3		nnon 6724	. . . . . . . . 9 |- ( x e. om -> x e. On )
4		elhf2.1	. . . . . . . . . 10 |- A e. _V
5	4	rankr1a 8332	. . . . . . . . 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 472	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) <-> ( rank ` A ) e. x ) )
8		elnn 6728	. . . . . . . . 9 |- ( ( ( rank ` A ) e. x /\ x e. om ) -> ( rank ` A ) e. om )
9	8	expcom 441	. . . . . . . 8 |- ( x e. om -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
10	9	adantl 472	. . . . . . 7 |- ( ( om e. On /\ x e. om ) -> ( ( rank ` A ) e. x -> ( rank ` A ) e. om ) )
11	7, 10	sylbid 223	. . . . . 6 |- ( ( om e. On /\ x e. om ) -> ( A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
12	11	rexlimdva 2890	. . . . 5 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) -> ( rank ` A ) e. om ) )
13		peano2 6739	. . . . . . . 8 |- ( ( rank ` A ) e. om -> suc ( rank ` A ) e. om )
14	13	adantr 471	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> suc ( rank ` A ) e. om )
15		r1rankid 8355	. . . . . . . . . 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 3968	. . . . . . . . 9 |- ( A e. ~P ( R1 ` ( rank ` A ) ) <-> A C_ ( R1 ` ( rank ` A ) ) )
18	16, 17	sylibr 217	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ~P ( R1 ` ( rank ` A ) ) )
19		nnon 6724	. . . . . . . . . 10 |- ( ( rank ` A ) e. om -> ( rank ` A ) e. On )
20		r1suc 8266	. . . . . . . . . 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 471	. . . . . . . 8 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
23	18, 22	eleqtrrd 2542	. . . . . . 7 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
24		fveq2 5887	. . . . . . . . 9 |- ( x = suc ( rank ` A ) -> ( R1 ` x ) = ( R1 ` suc ( rank ` A ) ) )
25	24	eleq2d 2524	. . . . . . . 8 |- ( x = suc ( rank ` A ) -> ( A e. ( R1 ` x ) <-> A e. ( R1 ` suc ( rank ` A ) ) ) )
26	25	rspcev 3161	. . . . . . 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 671	. . . . . 6 |- ( ( ( rank ` A ) e. om /\ om e. On ) -> E. x e. om A e. ( R1 ` x ) )
28	27	expcom 441	. . . . 5 |- ( om e. On -> ( ( rank ` A ) e. om -> E. x e. om A e. ( R1 ` x ) ) )
29	12, 28	impbid 195	. . . 4 |- ( om e. On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
30	4	tz9.13 8287	. . . . . 6 |- E. x e. On A e. ( R1 ` x )
31		rankon 8291	. . . . . 6 |- ( rank ` A ) e. On
32	30, 31	2th 247	. . . . 5 |- ( E. x e. On A e. ( R1 ` x ) <-> ( rank ` A ) e. On )
33		rexeq 2999	. . . . . 6 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> E. x e. On A e. ( R1 ` x ) ) )
34		eleq2 2528	. . . . . 6 |- ( om = On -> ( ( rank ` A ) e. om <-> ( rank ` A ) e. On ) )
35	33, 34	bibi12d 327	. . . . 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 241	. . . 4 |- ( om = On -> ( E. x e. om A e. ( R1 ` x ) <-> ( rank ` A ) e. om ) )
37	29, 36	jaoi 385	. . 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 257	1 |- ( A e. Hf <-> ( rank ` A ) e. om )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1454   e. wcel 1897   E.wrex 2749   _Vcvv 3056   C_ wss 3415   ~Pcpw 3962   Oncon0 5441   suc csuc 5443   ` cfv 5600   omcom 6718   R1cr1 8258   rankcrnk 8259   Hf chf 30987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-reg 8132  ax-inf2 8171

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-r1 8260  df-rank 8261  df-hf 30988

This theorem is referenced by:  elhf2g  30991  hfsn  30994