Theorem r1omhfb 35486

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. (Contributed by BTernaryTau, 24-Jan-2026.)

Assertion

Ref	Expression
r1omhfb	⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

Distinct variable group:   𝑥,𝐻,𝑦

Proof of Theorem r1omhfb

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

Step	Hyp	Ref	Expression
1		r1omhf 35478	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
2		eleq2w2 2759	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ 𝑥 ∈ ∪ (𝑅1 “ ω)))
3		eleq2w2 2759	. . . . . . 7 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑦 ∈ 𝐻 ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	3	ralbidv 3188	. . . . . 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 1957	. 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 1841	. . . 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 1841	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
14	13	ralrid 3087	. . . . . 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 1841	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
20		df-ss 3923	. . . . . 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		trssfir1om 35485	. . . . 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 1841	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
27		eleq1w 2846	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ 𝑤 ∈ ∪ (𝑅1 “ ω)))
28		eleq1w 2846	. . . . . . . 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 3839	. . . . . . 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 35478	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω)))
33		ralim 3105	. . . . . . . . . 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 2846	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ Fin ↔ 𝑧 ∈ Fin))
37		eleq1w 2846	. . . . . . . . . . . . 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 3340	. . . . . . . . . . 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 2846	. . . . . . . . . 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 2018	. . . . . . . 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	setinds2 9721	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻))
48	47	ssrdv 3944	. . . 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 3955	. 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 1568   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906  ∪ cuni 4873  Tr wtr 5219   “ cima 5666  ωcom 7863  Fincfn 8944  𝑅₁cr1 9735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-reg 9555  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-en 8945  df-dom 8946  df-fin 8948  df-r1 9737  df-rank 9738

This theorem is referenced by: (None)