Theorem r1omhfb 35372

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 35366	. . . 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 5210	. . . . . 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 3921	. . . . . 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		trssfir1om 35371	. . . . 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 3838	. . . . . . 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 35366	. . . . . . . . 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	setinds2 9703	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) → (𝑧 ∈ ∪ (𝑅1 “ ω) → 𝑧 ∈ 𝐻))
48	47	ssrdv 3942	. . . 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 3953	. 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 3904  ∪ cuni 4864  Tr wtr 5206   “ cima 5648  ωcom 7842  Fincfn 8923  𝑅₁cr1 9717

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-reg 9537  ax-inf2 9593

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 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-en 8924  df-dom 8925  df-fin 8927  df-r1 9719  df-rank 9720

This theorem is referenced by: (None)