Theorem mh-infprim2bi 37002

Description: Shortest possible axiom of infinity in primitive symbols not requiring ax-reg 9553. Deriving ax-inf 9606 or ax-inf2 9609 from this axiom requires ax-ext 2733 and ax-rep 5237, see mh-inf3sn 36997 and inf0 9589. (Contributed by Matthew House, 13-Apr-2026.)

Assertion

Ref	Expression
mh-infprim2bi	⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem mh-infprim2bi

Step	Hyp	Ref	Expression
1		sneq 4598	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
2	1	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ({𝑦} ∈ 𝑥 ↔ {𝑧} ∈ 𝑥))
3	2	cbvralvw 3241	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 {𝑧} ∈ 𝑥)
4		df-ral 3078	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 {𝑧} ∈ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥))
5	3, 4	bitri 278	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥))
6	5	anbi2i 634	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ (∅ ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)))
7		pwin 5552	. . . . . . . . 9 ⊢ 𝒫 ({𝑧} ∩ 𝑥) = (𝒫 {𝑧} ∩ 𝒫 𝑥)
8	7	raleqi 3319	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥)𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ (𝒫 {𝑧} ∩ 𝒫 𝑥)𝑦 ∈ 𝑥)
9		ralin 4201	. . . . . . . 8 ⊢ (∀𝑦 ∈ (𝒫 {𝑧} ∩ 𝒫 𝑥)𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝒫 {𝑧} (𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥))
10		pwsn 4864	. . . . . . . . 9 ⊢ 𝒫 {𝑧} = {∅, {𝑧}}
11	10	raleqi 3319	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝒫 {𝑧} (𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ {∅, {𝑧}} (𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥))
12	8, 9, 11	3bitrri 301	. . . . . . 7 ⊢ (∀𝑦 ∈ {∅, {𝑧}} (𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥)𝑦 ∈ 𝑥)
13		0ex 5269	. . . . . . . 8 ⊢ ∅ ∈ V
14		vsnex 5406	. . . . . . . 8 ⊢ {𝑧} ∈ V
15		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝒫 𝑥 ↔ ∅ ∈ 𝒫 𝑥))
16		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
17	15, 16	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ((𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ (∅ ∈ 𝒫 𝑥 → ∅ ∈ 𝑥)))
18		0elpw 5326	. . . . . . . . . 10 ⊢ ∅ ∈ 𝒫 𝑥
19	18	a1bi 365	. . . . . . . . 9 ⊢ (∅ ∈ 𝑥 ↔ (∅ ∈ 𝒫 𝑥 → ∅ ∈ 𝑥))
20	17, 19	bitr4di 292	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ ∅ ∈ 𝑥))
21		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = {𝑧} → (𝑦 ∈ 𝒫 𝑥 ↔ {𝑧} ∈ 𝒫 𝑥))
22		vex 3457	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
23	22	snelpw 5426	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 ↔ {𝑧} ∈ 𝒫 𝑥)
24	21, 23	bitr4di 292	. . . . . . . . 9 ⊢ (𝑦 = {𝑧} → (𝑦 ∈ 𝒫 𝑥 ↔ 𝑧 ∈ 𝑥))
25		eleq1 2849	. . . . . . . . 9 ⊢ (𝑦 = {𝑧} → (𝑦 ∈ 𝑥 ↔ {𝑧} ∈ 𝑥))
26	24, 25	imbi12d 347	. . . . . . . 8 ⊢ (𝑦 = {𝑧} → ((𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ (𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)))
27	13, 14, 20, 26	ralpr 4665	. . . . . . 7 ⊢ (∀𝑦 ∈ {∅, {𝑧}} (𝑦 ∈ 𝒫 𝑥 → 𝑦 ∈ 𝑥) ↔ (∅ ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)))
28		df-ral 3078	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥)𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥))
29	12, 27, 28	3bitr3i 304	. . . . . 6 ⊢ ((∅ ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)) ↔ ∀𝑦(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥))
30	29	albii 1847	. . . . 5 ⊢ (∀𝑧(∅ ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)) ↔ ∀𝑧∀𝑦(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥))
31		19.28v 2024	. . . . 5 ⊢ (∀𝑧(∅ ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)) ↔ (∅ ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)))
32		sneq 4598	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
33	32	ineq1d 4171	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ({𝑧} ∩ 𝑥) = ({𝑤} ∩ 𝑥))
34	33	pweqd 4578	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → 𝒫 ({𝑧} ∩ 𝑥) = 𝒫 ({𝑤} ∩ 𝑥))
35	34	eleq2d 2847	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) ↔ 𝑦 ∈ 𝒫 ({𝑤} ∩ 𝑥)))
36	35	imbi1d 344	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝒫 ({𝑤} ∩ 𝑥) → 𝑦 ∈ 𝑥)))
37		eleq1w 2844	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) ↔ 𝑤 ∈ 𝒫 ({𝑧} ∩ 𝑥)))
38		elequ1 2148	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
39	37, 38	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥) ↔ (𝑤 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑤 ∈ 𝑥)))
40	36, 39	alcomw 2073	. . . . 5 ⊢ (∀𝑧∀𝑦(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥) ↔ ∀𝑦∀𝑧(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥))
41	30, 31, 40	3bitr3i 304	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → {𝑧} ∈ 𝑥)) ↔ ∀𝑦∀𝑧(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥))
42		velpw 4566	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) ↔ 𝑦 ⊆ ({𝑧} ∩ 𝑥))
43		df-ss 3921	. . . . . . 7 ⊢ (𝑦 ⊆ ({𝑧} ∩ 𝑥) ↔ ∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ ({𝑧} ∩ 𝑥)))
44		elin 3920	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑧} ∩ 𝑥) ↔ (𝑤 ∈ {𝑧} ∧ 𝑤 ∈ 𝑥))
45		velsn 4604	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
46	45	anbi2ci 636	. . . . . . . . . 10 ⊢ ((𝑤 ∈ {𝑧} ∧ 𝑤 ∈ 𝑥) ↔ (𝑤 ∈ 𝑥 ∧ 𝑤 = 𝑧))
47		df-an 401	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑤 = 𝑧) ↔ ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧))
48	44, 46, 47	3bitri 300	. . . . . . . . 9 ⊢ (𝑤 ∈ ({𝑧} ∩ 𝑥) ↔ ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧))
49	48	imbi2i 339	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦 → 𝑤 ∈ ({𝑧} ∩ 𝑥)) ↔ (𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)))
50	49	albii 1847	. . . . . . 7 ⊢ (∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ ({𝑧} ∩ 𝑥)) ↔ ∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)))
51	42, 43, 50	3bitri 300	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) ↔ ∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)))
52	51	imbi1i 352	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥) ↔ (∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))
53	52	2albii 1848	. . . 4 ⊢ (∀𝑦∀𝑧(𝑦 ∈ 𝒫 ({𝑧} ∩ 𝑥) → 𝑦 ∈ 𝑥) ↔ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))
54	6, 41, 53	3bitri 300	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))
55	54	exbii 1876	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ ∃𝑥∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))
56		df-ex 1808	. 2 ⊢ (∃𝑥∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))
57	55, 56	bitri 278	1 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∀wral 3077   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588  {cpr 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4563  df-sn 4589  df-pr 4591

This theorem is referenced by: (None)