Theorem elhf2 9917

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 9914	. 2 ⊢ (𝐴 ∈ HF ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
2		omon 7893	. . 3 ⊢ (ω ∈ On ∨ ω = On)
3		nnon 7887	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
4		elhf2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5	4	rankr1a 9855	. . . . . . . . 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 7892	. . . . . . . . 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 3163	. . . . 5 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
13		peano2 7905	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
14	13	adantr 486	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15		r1rankid 9882	. . . . . . . . . 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 4560	. . . . . . . . 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 7887	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20		r1suc 9784	. . . . . . . . . 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 2863	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24		fveq2 6890	. . . . . . . . 9 ⊢ (𝑥 = suc (rank‘𝐴) → (𝑅1‘𝑥) = (𝑅1‘suc (rank‘𝐴)))
25	24	eleq2d 2846	. . . . . . . 8 ⊢ (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
26	25	rspcev 3576	. . . . . . 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 9805	. . . . . 6 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
31		rankon 9810	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
32	30, 31	2th 267	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)
33		rexeq 3315	. . . . . 6 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
34		eleq2 2849	. . . . . 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 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896  𝒫 cpw 4556  Oncon0 6365  suc csuc 6367  ‘cfv 6543  ωcom 7881  𝑅₁cr1 9773  rankcrnk 9774   HF chf 9911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-reg 9593  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-r1 9775  df-rank 9776  df-hf 9912

This theorem is used by:  elhf2g  9918  hfsnOLD  9928