Theorem elhf2 36376

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 36375	. 2 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
2		omon 7823	. . 3 ⊢ (ω ∈ On ∨ ω = On)
3		nnon 7817	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
4		elhf2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5	4	rankr1a 9754	. . . . . . . . 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 7822	. . . . . . . . 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 3139	. . . . 5 ⊢ (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) → (rank‘𝐴) ∈ ω))
13		peano2 7835	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
14	13	adantr 480	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15		r1rankid 9777	. . . . . . . . . 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 4546	. . . . . . . . 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 7817	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20		r1suc 9688	. . . . . . . . . 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 2840	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24		fveq2 6835	. . . . . . . . 9 ⊢ (𝑥 = suc (rank‘𝐴) → (𝑅1‘𝑥) = (𝑅1‘suc (rank‘𝐴)))
25	24	eleq2d 2823	. . . . . . . 8 ⊢ (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
26	25	rspcev 3565	. . . . . . 7 ⊢ ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
27	14, 23, 26	syl2anc 585	. . . . . 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 9709	. . . . . 6 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
31		rankon 9713	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
32	30, 31	2th 264	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) ↔ (rank‘𝐴) ∈ On)
33		rexeq 3292	. . . . . 6 ⊢ (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
34		eleq2 2826	. . . . . 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 858	. . 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 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  Oncon0 6318  suc csuc 6320  ‘cfv 6493  ωcom 7811  𝑅₁cr1 9680  rankcrnk 9681   Hf chf 36373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-reg 9501  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-r1 9682  df-rank 9683  df-hf 36374

This theorem is referenced by:  elhf2g  36377  hfsn  36380