Theorem r1omhfbregs 35469

Description: The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of r1omhfb 35444 replaces setinds2 9716 with setinds2regs 35463 and trssfir1om 35443 with trssfir1omregs 35468. (Contributed by BTernaryTau, 21-Jan-2026.)

Assertion

Ref	Expression
r1omhfbregs	⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

Distinct variable group:   𝑥,𝐻,𝑦

Proof of Theorem r1omhfbregs

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1omhf 35438	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
2		eleq2w2 2765	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ 𝑥 ∈ ∪ (𝑅1 “ ω)))
3		eleq2w2 2765	. . . . . . 7 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑦 ∈ 𝐻 ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	3	ralbidv 3194	. . . . . 6 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
5	4	anbi2d 641	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω))))
6	2, 5	bibi12d 348	. . . 4 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) ↔ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))))
7	1, 6	mpbiri 261	. . 3 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
8	7	alrimiv 1954	. 2 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
9		biimp 218	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
10	9	alimi 1838	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
11		simpr 489	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
12	11	imim2i 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
13	12	alimi 1838	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
14	13	ralrid 3093	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
15		dftr5 5223	. . . . . 6 ⊢ (Tr 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
16	14, 15	sylibr 237	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → Tr 𝐻)
17		simpl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ Fin)
18	17	imim2i 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
19	18	alimi 1838	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
20		df-ss 3930	. . . . . 6 ⊢ (𝐻 ⊆ Fin ↔ ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
21	19, 20	sylibr 237	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ Fin)
22		trssfir1omregs 35468	. . . . 5 ⊢ ((Tr 𝐻 ∧ 𝐻 ⊆ Fin) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
23	16, 21, 22	syl2anc 595	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
24	10, 23	syl 18	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
25		biimpr 223	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
26	25	alimi 1838	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
27		eleq1w 2852	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ 𝑤 ∈ ∪ (𝑅1 “ ω)))
28		eleq1w 2852	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
29	27, 28	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻) ↔ (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)))
30	29	imbi2d 343	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻)) ↔ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻))))
31		ra4v 3847	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝑧 (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)) → (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → ∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)))
32		r1omhf 35438	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω)))
33		ralim 3111	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω) → ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
34	33	anim2d 623	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω)) → (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
35	32, 34	biimtrid 245	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
36		eleq1w 2852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ Fin ↔ 𝑧 ∈ Fin))
37		eleq1w 2852	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
38	37	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
39		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
40	38, 39	cbvraldva2 3347	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
41	36, 40	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
42		eleq1w 2852	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐻 ↔ 𝑧 ∈ 𝐻))
43	41, 42	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) ↔ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻) → 𝑧 ∈ 𝐻)))
44	43	spvv 2015	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻) → 𝑧 ∈ 𝐻))
45	35, 44	syl9r 79	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻)))
46	31, 45	sylcom 31	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑧 (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)) → (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻)))
47	30, 46	setinds2regs 35463	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻))
48	47	ssrdv 3951	. . . 4 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → ∪ (𝑅1 “ ω) ⊆ 𝐻)
49	26, 48	syl 18	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∪ (𝑅1 “ ω) ⊆ 𝐻)
50	24, 49	eqssd 3962	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 = ∪ (𝑅1 “ ω))
51	8, 50	impbii 212	1 ⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ∪ cuni 4873  Tr wtr 5219   “ cima 5662  ωcom 7858  Fincfn 8939  𝑅₁cr1 9730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-regs 35458

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-en 8940  df-dom 8941  df-fin 8943  df-r1 9732  df-rank 9733

This theorem is referenced by: (None)