Theorem elhf2 36219

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 36218	. 2 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
2		omon 7808	. . 3 ⊢ (ω ∈ On ∨ ω = On)
3		nnon 7802	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
4		elhf2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5	4	rankr1a 9729	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8		elnn 7807	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
9	8	expcom 413	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
10	9	adantl 481	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
11	7, 10	sylbid 240	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
12	11	rexlimdva 3133	. . . . 5 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
13		peano2 7820	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
14	13	adantr 480	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15		r1rankid 9752	. . . . . . . . . 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 4551	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
18	16, 17	sylibr 234	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19		nnon 7802	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20		r1suc 9663	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
23	18, 22	eleqtrrd 2834	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24		fveq2 6822	. . . . . . . . 9 ⊢ (𝑥 = suc (rank‘𝐴) → (𝑅1‘𝑥) = (𝑅1‘suc (rank‘𝐴)))
25	24	eleq2d 2817	. . . . . . . 8 ⊢ (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
26	25	rspcev 3572	. . . . . . 7 ⊢ ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
27	14, 23, 26	syl2anc 584	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
28	27	expcom 413	. . . . 5 ⊢ (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)))
29	12, 28	impbid 212	. . . 4 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
30	4	tz9.13 9684	. . . . . 6 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
31		rankon 9688	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
32	30, 31	2th 264	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)
33		rexeq 3288	. . . . . 6 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
34		eleq2 2820	. . . . . 6 ⊢ (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
35	33, 34	bibi12d 345	. . . . 5 ⊢ (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)))
36	32, 35	mpbiri 258	. . . 4 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ ω))
37	29, 36	jaoi 857	. . 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 275	1 ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  Oncon0 6306  suc csuc 6308  ‘cfv 6481  ωcom 7796  𝑅₁cr1 9655  rankcrnk 9656   Hf chf 36216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-reg 9478  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-r1 9657  df-rank 9658  df-hf 36217

This theorem is referenced by:  elhf2g  36220  hfsn  36223