Theorem nregmodel 45944

Description: The Axiom of Regularity ax-reg 9564 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 45932 through permac8prim 45941), 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 45920 through wfac8prim 45929), 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 45943	. . . . . . . 8 ⊢ (𝑦𝑅∅ ↔ 𝑦 ∈ {∅})
9	5, 8	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦 ∈ {∅}))
10	9	exbidv 1954	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑦 ∈ {∅}))
11		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧𝑅∅))
12	6, 7	nregmodellem 45943	. . . . . . . . . . . 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 1953	. . . . . . . 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 1954	. . . . . 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 1953	. . . . . 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 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 3895   ∪ cun 3896  ∅c0 4278  {csn 4583  {cpr 4585  ⟨cop 4589   class class class wbr 5102   I cid 5541   E cep 5546  ^◡ccnv 5646   ↾ cres 5649   ∘ ccom 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535

This theorem is used by: (None)