Theorem mh-infprim3bi 36789

Description: An axiom of infinity in primitive symbols not requiring ax-reg 9501. This version of the axiom was designed by Stefan O'Rear for his zf2.nql program, see https://github.com/sorear/metamath-turing-machines 9501. It directly implies ax-inf 9554, but deriving ax-inf2 9557 requires ax-ext 2713 and ax-rep 5201, see mh-inf3sn 36783. (Contributed by Matthew House, 13-Apr-2026.)

Assertion

Ref	Expression
mh-infprim3bi	⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ ¬ ∀𝑦 ¬ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem mh-infprim3bi

Step	Hyp	Ref	Expression
1		sneq 4567	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
2	1	eleq1d 2826	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ({𝑧} ∈ 𝑦 ↔ {𝑥} ∈ 𝑦))
3	2	cbvralvw 3219	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦 ↔ ∀𝑥 ∈ 𝑦 {𝑥} ∈ 𝑦)
4		dfclel 2817	. . . . . . . 8 ⊢ ({𝑥} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑧 ∈ 𝑦))
5		dfcleq 2734	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}))
6		velsn 4573	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
7	6	bibi2i 339	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}) ↔ (𝑦 ∈ 𝑧 ↔ 𝑦 = 𝑥))
8		dfbi1 215	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑧 ↔ 𝑦 = 𝑥) ↔ ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))
9	7, 8	bitri 277	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}) ↔ ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))
10	9	albii 1827	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))
11	5, 10	bitri 277	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑥} ↔ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))
12	11	anbi2ci 632	. . . . . . . . . 10 ⊢ ((𝑧 = {𝑥} ∧ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑦 ∧ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
13		df-an 398	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))) ↔ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
14	12, 13	bitri 277	. . . . . . . . 9 ⊢ ((𝑧 = {𝑥} ∧ 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
15	14	exbii 1856	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = {𝑥} ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑧 ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
16		df-ex 1788	. . . . . . . 8 ⊢ (∃𝑧 ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))) ↔ ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
17	4, 15, 16	3bitri 299	. . . . . . 7 ⊢ ({𝑥} ∈ 𝑦 ↔ ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
18	17	ralbii 3087	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑦 {𝑥} ∈ 𝑦 ↔ ∀𝑥 ∈ 𝑦 ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))
19		df-ral 3056	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑦 ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))))
20	3, 18, 19	3bitri 299	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦 ↔ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧)))))
21	20	anbi2i 630	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))
22		df-an 398	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))) ↔ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))
23	21, 22	bitri 277	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))
24	23	exbii 1856	. 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ ∃𝑦 ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))
25		df-ex 1788	. 2 ⊢ (∃𝑦 ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))) ↔ ¬ ∀𝑦 ¬ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))
26	24, 25	bitri 277	1 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ ¬ ∀𝑦 ¬ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055  {csn 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-v 3435  df-sn 4558

This theorem is referenced by: (None)