Theorem elhf2 36706

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 36705	. 2 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
2		omon 7880	. . 3 ⊢ (ω ∈ On ∨ ω = On)
3		nnon 7874	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
4		elhf2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5	4	rankr1a 9815	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
6	3, 5	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
7	6	adantl 487	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8		elnn 7879	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
9	8	expcom 419	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
10	9	adantl 487	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
11	7, 10	sylbid 243	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
12	11	rexlimdva 3168	. . . . 5 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
13		peano2 7892	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
14	13	adantr 486	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15		r1rankid 9838	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
16	4, 15	mp1i 14	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
17	4	elpw 4568	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
18	16, 17	sylibr 237	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19		nnon 7874	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20		r1suc 9749	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ On → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
21	19, 20	syl 18	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∈ ω → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
22	21	adantr 486	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
23	18, 22	eleqtrrd 2868	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = suc (rank‘𝐴) → (𝑅1‘𝑥) = (𝑅1‘suc (rank‘𝐴)))
25	24	eleq2d 2851	. . . . . . . 8 ⊢ (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
26	25	rspcev 3583	. . . . . . 7 ⊢ ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
27	14, 23, 26	syl2anc 596	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
28	27	expcom 419	. . . . 5 ⊢ (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)))
29	12, 28	impbid 215	. . . 4 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
30	4	tz9.13 9770	. . . . . 6 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
31		rankon 9774	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
32	30, 31	2th 267	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)
33		rexeq 3321	. . . . . 6 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
34		eleq2 2854	. . . . . 6 ⊢ (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
35	33, 34	bibi12d 348	. . . . 5 ⊢ (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)))
36	32, 35	mpbiri 261	. . . 4 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
37	29, 36	jaoi 871	. . 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 278	1 ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  𝒫 cpw 4564  Oncon0 6364  suc csuc 6366  ‘cfv 6540  ωcom 7868  𝑅₁cr1 9741  rankcrnk 9742   Hf chf 36703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-r1 9743  df-rank 9744  df-hf 36704

This theorem is used by:  elhf2g  36707  hfsn  36710