Theorem nregmodel 45017

Description: The Axiom of Regularity ax-reg 9611 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 45005 through permac8prim 45014), 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 44993 through wfac8prim 45002), 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 5282	. . 3 ⊢ ∅ ∈ V
2	1	snid 4643	. 2 ⊢ ∅ ∈ {∅}
3		eleq1 2823	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 ∈ {∅} ↔ ∅ ∈ {∅}))
4	1, 3, 2	ceqsexv2d 3517	. . . 4 ⊢ ∃𝑦 𝑦 ∈ {∅}
5		breq2 5128	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦𝑅∅))
6		nregmodel.1	. . . . . . . . 9 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		nregmodel.2	. . . . . . . . 9 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	6, 7	nregmodellem 45016	. . . . . . . 8 ⊢ (𝑦𝑅∅ ↔ 𝑦 ∈ {∅})
9	5, 8	bitrdi 287	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦 ∈ {∅}))
10	9	exbidv 1921	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑦 ∈ {∅}))
11		breq2 5128	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧𝑅∅))
12	6, 7	nregmodellem 45016	. . . . . . . . . . . 12 ⊢ (𝑧𝑅∅ ↔ 𝑧 ∈ {∅})
13	11, 12	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧 ∈ {∅}))
14	13	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑧 ∈ {∅}))
15	14	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥) ↔ (𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
16	15	albidv 1920	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
17	9, 16	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)) ↔ (𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
18	17	exbidv 1921	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)) ↔ ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
19	10, 18	imbi12d 344	. . . . 5 ⊢ (𝑥 = ∅ → ((∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) ↔ (∃𝑦 𝑦 ∈ {∅} → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))))
20	1, 19	spcv 3589	. . . 4 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → (∃𝑦 𝑦 ∈ {∅} → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
21	4, 20	mpi 20	. . 3 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
22		df-ral 3053	. . . . 5 ⊢ (∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅} ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
23		breq2 5128	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧𝑅𝑦 ↔ 𝑧𝑅∅))
24	23, 12	bitrdi 287	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑧𝑅𝑦 ↔ 𝑧 ∈ {∅}))
25	24	imbi1d 341	. . . . . . 7 ⊢ (𝑦 = ∅ → ((𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ (𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅})))
26	25	albidv 1920	. . . . . 6 ⊢ (𝑦 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅})))
27	1, 26	rexsn 4663	. . . . 5 ⊢ (∃𝑦 ∈ {∅}∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
28		df-rex 3062	. . . . 5 ⊢ (∃𝑦 ∈ {∅}∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
29	22, 27, 28	3bitr2ri 300	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})) ↔ ∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅})
30		eleq1 2823	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑧 ∈ {∅} ↔ ∅ ∈ {∅}))
31	30	notbid 318	. . . . 5 ⊢ (𝑧 = ∅ → (¬ 𝑧 ∈ {∅} ↔ ¬ ∅ ∈ {∅}))
32	1, 31	ralsn 4662	. . . 4 ⊢ (∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅} ↔ ¬ ∅ ∈ {∅})
33	29, 32	bitri 275	. . 3 ⊢ (∃𝑦(𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})) ↔ ¬ ∅ ∈ {∅})
34	21, 33	sylib 218	. 2 ⊢ (∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥))) → ¬ ∅ ∈ {∅})
35	2, 34	mt2 200	1 ⊢ ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929  ∅c0 4313  {csn 4606  {cpr 4608  ⟨cop 4612   class class class wbr 5124   I cid 5552   E cep 5557  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-eprel 5558  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)