Theorem nregmodel 46083

Description: The Axiom of Regularity ax-reg 9587 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 46071 through permac8prim 46080), we can conclude that Regularity does not follow from those axioms, assuming ZFC is consistent. (If we could prove Regularity from the other axioms, we could prove it in the permutation model and thus obtain a contradiction with this theorem.) Since we also know that Regularity is consistent with the other axioms (wfaxext 46059 through wfac8prim 46068), Regularity is neither provable nor disprovable from the other axioms; i.e., it is independent of them. (Contributed by Eric Schmidt, 16-Nov-2025.)

Hypotheses

Ref	Expression
nregmodel.1	⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
nregmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
nregmodel	⊢ ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))

Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑅

Allowed substitution hints:   𝑅(𝑦, 𝑧)   𝐹(𝑥, 𝑦, 𝑧)

Proof of Theorem nregmodel

Step	Hyp	Ref	Expression
1		0ex 5260	. . 3 ⊢ ∅ ∈ V
2	1	snid 4622	. 2 ⊢ ∅ ∈ {∅}
3		eleq1 2848	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 ∈ {∅} ↔ ∅ ∈ {∅}))
4	1, 3, 2	ceqsexv2d 3499	. . . 4 ⊢ ∃𝑦 𝑦 ∈ {∅}
5		breq2 5106	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦𝑅∅))
6		nregmodel.1	. . . . . . . . 9 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		nregmodel.2	. . . . . . . . 9 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	6, 7	nregmodellem 46082	. . . . . . . 8 ⊢ (𝑦𝑅∅ ↔ 𝑦 ∈ {∅})
9	5, 8	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦 ∈ {∅}))
10	9	exbidv 1953	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑦 ∈ {∅}))
11		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧𝑅∅))
12	6, 7	nregmodellem 46082	. . . . . . . . . . . 12 ⊢ (𝑧𝑅∅ ↔ 𝑧 ∈ {∅})
13	11, 12	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧 ∈ {∅}))
14	13	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑧 ∈ {∅}))
15	14	imbi2d 343	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥) ↔ (𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
16	15	albidv 1952	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
17	9, 16	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)) ↔ (𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
18	17	exbidv 1953	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)) ↔ ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
19	10, 18	imbi12d 347	. . . . 5 ⊢ (𝑥 = ∅ → ((∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) ↔ (∃𝑦 𝑦 ∈ {∅} → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))))
20	1, 19	spcv 3559	. . . 4 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → (∃𝑦 𝑦 ∈ {∅} → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
21	4, 20	mpi 21	. . 3 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
22		df-ral 3077	. . . . 5 ⊢ (∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅} ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
23		breq2 5106	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧𝑅𝑦 ↔ 𝑧𝑅∅))
24	23, 12	bitrdi 290	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑧𝑅𝑦 ↔ 𝑧 ∈ {∅}))
25	24	imbi1d 344	. . . . . . 7 ⊢ (𝑦 = ∅ → ((𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ (𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅})))
26	25	albidv 1952	. . . . . 6 ⊢ (𝑦 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅})))
27	1, 26	rexsn 4642	. . . . 5 ⊢ (∃𝑦 ∈ {∅}∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
28		df-rex 3087	. . . . 5 ⊢ (∃𝑦 ∈ {∅}∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
29	22, 27, 28	3bitr2ri 303	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})) ↔ ∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅})
30		eleq1 2848	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑧 ∈ {∅} ↔ ∅ ∈ {∅}))
31	30	notbid 321	. . . . 5 ⊢ (𝑧 = ∅ → (¬ 𝑧 ∈ {∅} ↔ ¬ ∅ ∈ {∅}))
32	1, 31	ralsn 4641	. . . 4 ⊢ (∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅} ↔ ¬ ∅ ∈ {∅})
33	29, 32	bitri 278	. . 3 ⊢ (∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})) ↔ ¬ ∅ ∈ {∅})
34	21, 33	sylib 221	. 2 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → ¬ ∅ ∈ {∅})
35	2, 34	mt2 203	1 ⊢ ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4583  {cpr 4585  ⟨cop 4589   class class class wbr 5102   I cid 5544   E cep 5549  ^◡ccnv 5649   ↾ cres 5652   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by: (None)