Theorem r1omhfb 35272

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 35265	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
2		eleq2w2 2733	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ 𝑥 ∈ ∪ (𝑅1 “ ω)))
3		eleq2w2 2733	. . . . . . 7 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑦 ∈ 𝐻 ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	3	ralbidv 3161	. . . . . 6 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
5	4	anbi2d 631	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω))))
6	2, 5	bibi12d 345	. . . 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 258	. . 3 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
8	7	alrimiv 1929	. 2 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
9		biimp 215	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
10	9	alimi 1813	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
11		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
12	11	imim2i 16	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
13	12	alimi 1813	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
14	13	ralrid 3060	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
15		dftr5 5197	. . . . . 6 ⊢ (Tr 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
16	14, 15	sylibr 234	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → Tr 𝐻)
17		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ Fin)
18	17	imim2i 16	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
19	18	alimi 1813	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
20		df-ss 3907	. . . . . 6 ⊢ (𝐻 ⊆ Fin ↔ ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
21	19, 20	sylibr 234	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ Fin)
22		trssfir1om 35271	. . . . 5 ⊢ ((Tr 𝐻 ∧ 𝐻 ⊆ Fin) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
23	16, 21, 22	syl2anc 585	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
24	10, 23	syl 17	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
25		biimpr 220	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
26	25	alimi 1813	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
27		eleq1w 2820	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ 𝑤 ∈ ∪ (𝑅1 “ ω)))
28		eleq1w 2820	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
29	27, 28	imbi12d 344	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻) ↔ (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)))
30	29	imbi2d 340	. . . . . 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 3824	. . . . . . 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 35265	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω)))
33		ralim 3078	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω) → ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
34	33	anim2d 613	. . . . . . . . 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 242	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
36		eleq1w 2820	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ Fin ↔ 𝑧 ∈ Fin))
37		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
39		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
40	38, 39	cbvraldva2 3314	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
41	36, 40	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
42		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐻 ↔ 𝑧 ∈ 𝐻))
43	41, 42	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) ↔ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻) → 𝑧 ∈ 𝐻)))
44	43	spvv 1990	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻) → 𝑧 ∈ 𝐻))
45	35, 44	syl9r 78	. . . . . . 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 30	. . . . . 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 9663	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻))
48	47	ssrdv 3928	. . . 4 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → ∪ (𝑅1 “ ω) ⊆ 𝐻)
49	26, 48	syl 17	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∪ (𝑅1 “ ω) ⊆ 𝐻)
50	24, 49	eqssd 3940	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 = ∪ (𝑅1 “ ω))
51	8, 50	impbii 209	1 ⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ∪ cuni 4851  Tr wtr 5193   “ cima 5627  ωcom 7810  Fincfn 8886  𝑅₁cr1 9677

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553

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 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-en 8887  df-dom 8888  df-fin 8890  df-r1 9679  df-rank 9680

This theorem is referenced by: (None)