Theorem elhf2 32816

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

Hypothesis

Ref	Expression
elhf2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elhf2	⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)

Proof of Theorem elhf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhf 32815	. 2 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
2		omon 7342	. . 3 ⊢ (ω ∈ On ∨ ω = On)
3		nnon 7337	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
4		elhf2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5	4	rankr1a 8983	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
7	6	adantl 475	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8		elnn 7341	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
9	8	expcom 404	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
10	9	adantl 475	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
11	7, 10	sylbid 232	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
12	11	rexlimdva 3240	. . . . 5 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
13		peano2 7352	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
14	13	adantr 474	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15		r1rankid 9006	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
16	4, 15	mp1i 13	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
17	4	elpw 4386	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
18	16, 17	sylibr 226	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19		nnon 7337	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20		r1suc 8917	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ On → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∈ ω → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
22	21	adantr 474	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
23	18, 22	eleqtrrd 2909	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24		fveq2 6437	. . . . . . . . 9 ⊢ (𝑥 = suc (rank‘𝐴) → (𝑅1‘𝑥) = (𝑅1‘suc (rank‘𝐴)))
25	24	eleq2d 2892	. . . . . . . 8 ⊢ (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
26	25	rspcev 3526	. . . . . . 7 ⊢ ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
27	14, 23, 26	syl2anc 579	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
28	27	expcom 404	. . . . 5 ⊢ (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)))
29	12, 28	impbid 204	. . . 4 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
30	4	tz9.13 8938	. . . . . 6 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
31		rankon 8942	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
32	30, 31	2th 256	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)
33		rexeq 3351	. . . . . 6 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
34		eleq2 2895	. . . . . 6 ⊢ (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
35	33, 34	bibi12d 337	. . . . 5 ⊢ (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)))
36	32, 35	mpbiri 250	. . . 4 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
37	29, 36	jaoi 888	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
38	2, 37	ax-mp 5	. 2 ⊢ (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω)
39	1, 38	bitri 267	1 ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4380  Oncon0 5967  suc csuc 5969  ‘cfv 6127  ωcom 7331  𝑅₁cr1 8909  rankcrnk 8910   Hf chf 32813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-reg 8773  ax-inf2 8822

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-r1 8911  df-rank 8912  df-hf 32814

This theorem is referenced by:  elhf2g  32817  hfsn  32820