Theorem r1omhfbregs 35394

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 35369 replaces setinds2 9700 with setinds2regs 35388 and trssfir1om 35368 with trssfir1omregs 35393. (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 35363	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
2		eleq2w2 2757	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ 𝑥 ∈ ∪ (𝑅1 “ ω)))
3		eleq2w2 2757	. . . . . . 7 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑦 ∈ 𝐻 ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	3	ralbidv 3184	. . . . . 6 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
5	4	anbi2d 639	. . . . 5 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω))))
6	2, 5	bibi12d 347	. . . 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 260	. . 3 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → (𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
8	7	alrimiv 1946	. 2 ⊢ (𝐻 = ∪ (𝑅1 “ ω) → ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
9		biimp 217	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → (𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
10	9	alimi 1830	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
11		simpr 488	. . . . . . . . 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 1830	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻))
14	13	ralrid 3083	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
15		dftr5 5208	. . . . . 6 ⊢ (Tr 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)
16	14, 15	sylibr 236	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → Tr 𝐻)
17		simpl 486	. . . . . . . 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 1830	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
20		df-ss 3919	. . . . . 6 ⊢ (𝐻 ⊆ Fin ↔ ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ Fin))
21	19, 20	sylibr 236	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐻 → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 ⊆ Fin)
22		trssfir1omregs 35393	. . . . 5 ⊢ ((Tr 𝐻 ∧ 𝐻 ⊆ Fin) → 𝐻 ⊆ ∪ (𝑅1 “ ω))
23	16, 21, 22	syl2anc 593	. . . 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 222	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
26	25	alimi 1830	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → ∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻))
27		eleq1w 2844	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ 𝑤 ∈ ∪ (𝑅1 “ ω)))
28		eleq1w 2844	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
29	27, 28	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻) ↔ (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻)))
30	29	imbi2d 342	. . . . . 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 3836	. . . . . . 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 35363	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ (𝑅1 “ ω) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω)))
33		ralim 3101	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (∀𝑤 ∈ 𝑧 𝑤 ∈ ∪ (𝑅1 “ ω) → ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
34	33	anim2d 621	. . . . . . . . 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 244	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑧 (𝑤 ∈ ∪ (𝑅1 “ ω) → 𝑤 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
36		eleq1w 2844	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ Fin ↔ 𝑧 ∈ Fin))
37		eleq1w 2844	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
38	37	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐻 ↔ 𝑤 ∈ 𝐻))
39		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
40	38, 39	cbvraldva2 3337	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻 ↔ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻))
41	36, 40	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) ↔ (𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻)))
42		eleq1w 2844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐻 ↔ 𝑧 ∈ 𝐻))
43	41, 42	imbi12d 346	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) ↔ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 𝑤 ∈ 𝐻) → 𝑧 ∈ 𝐻)))
44	43	spvv 2007	. . . . . . . 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	setinds2regs 35388	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻))
48	47	ssrdv 3940	. . . 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 3951	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)) → 𝐻 = ∪ (𝑅1 “ ω))
51	8, 50	impbii 211	1 ⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3902  ∪ cuni 4862  Tr wtr 5204   “ cima 5646  ωcom 7841  Fincfn 8921  𝑅₁cr1 9714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-regs 35383

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-en 8922  df-dom 8923  df-fin 8925  df-r1 9716  df-rank 9717

This theorem is referenced by: (None)