Theorem nregmodel 45696

Description: The Axiom of Regularity ax-reg 9553 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 45684 through permac8prim 45693), 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 45672 through wfac8prim 45681), 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 5269	. . 3 ⊢ ∅ ∈ V
2	1	snid 4627	. 2 ⊢ ∅ ∈ {∅}
3		eleq1 2849	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 ∈ {∅} ↔ ∅ ∈ {∅}))
4	1, 3, 2	ceqsexv2d 3502	. . . 4 ⊢ ∃𝑦 𝑦 ∈ {∅}
5		breq2 5112	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦𝑅∅))
6		nregmodel.1	. . . . . . . . 9 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		nregmodel.2	. . . . . . . . 9 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	6, 7	nregmodellem 45695	. . . . . . . 8 ⊢ (𝑦𝑅∅ ↔ 𝑦 ∈ {∅})
9	5, 8	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦𝑅𝑥 ↔ 𝑦 ∈ {∅}))
10	9	exbidv 1949	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑦 ∈ {∅}))
11		breq2 5112	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑧𝑅𝑥 ↔ 𝑧𝑅∅))
12	6, 7	nregmodellem 45695	. . . . . . . . . . . 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 1948	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅})))
17	9, 16	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)) ↔ (𝑦 ∈ {∅} ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}))))
18	17	exbidv 1949	. . . . . 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 3563	. . . 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 3078	. . . . 5 ⊢ (∀𝑧 ∈ {∅} ¬ 𝑧 ∈ {∅} ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
23		breq2 5112	. . . . . . . . 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 1948	. . . . . 6 ⊢ (𝑦 = ∅ → (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅})))
27	1, 26	rexsn 4647	. . . . 5 ⊢ (∃𝑦 ∈ {∅}∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ {∅}) ↔ ∀𝑧(𝑧 ∈ {∅} → ¬ 𝑧 ∈ {∅}))
28		df-rex 3088	. . . . 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 2849	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑧 ∈ {∅} ↔ ∅ ∈ {∅}))
31	30	notbid 321	. . . . 5 ⊢ (𝑧 = ∅ → (¬ 𝑧 ∈ {∅} ↔ ¬ ∅ ∈ {∅}))
32	1, 31	ralsn 4646	. . . 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902  ∅c0 4285  {csn 4588  {cpr 4590  ⟨cop 4594   class class class wbr 5108   I cid 5555   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   ∘ ccom 5665

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-10 2174  ax-11 2190  ax-12 2211  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-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  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)