Theorem fineqvinfep 35288

Description: A counterexample demonstrating that tz9.1 9644 does not hold when all sets are finite and an infinite descending ∈-chain exists. (Contributed by BTernaryTau, 18-Feb-2026.)

Hypothesis

Ref	Expression
fineqvinfep.1	⊢ 𝐴 = {(𝐹‘∅)}

Assertion

Ref	Expression
fineqvinfep	⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))

Distinct variable group:   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem fineqvinfep

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
2		eleq2 2826	. . . . 5 ⊢ (Fin = V → (𝑦 ∈ Fin ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 258	. . . 4 ⊢ (Fin = V → 𝑦 ∈ Fin)
4	3	3ad2ant1 1134	. . 3 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → 𝑦 ∈ Fin)
5		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝐹‘𝑤) = (𝐹‘∅))
6	5	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘∅) ∈ 𝑦))
7		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧))
8	7	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘𝑧) ∈ 𝑦))
9		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → (𝐹‘𝑤) = (𝐹‘suc 𝑧))
10	9	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘suc 𝑧) ∈ 𝑦))
11		simp2 1138	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝐴 ⊆ 𝑦)
12		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘∅) ∈ V
13	12	snid 4607	. . . . . . . . . . . . . 14 ⊢ (𝐹‘∅) ∈ {(𝐹‘∅)}
14		fineqvinfep.1	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {(𝐹‘∅)}
15	13, 14	eleqtrri 2836	. . . . . . . . . . . . 13 ⊢ (𝐹‘∅) ∈ 𝐴
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹‘∅) ∈ 𝐴)
17	11, 16	sseldd 3923	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹‘∅) ∈ 𝑦)
18		3simpb 1150	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ Tr 𝑦))
19		suceq 6386	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
20	19	fveq2d 6839	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
21		fveq2 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
22	20, 21	eleq12d 2831	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
23	22	rspcv 3561	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
24		trel 5201	. . . . . . . . . . . . . . . 16 ⊢ (Tr 𝑦 → (((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ∈ 𝑦) → (𝐹‘suc 𝑧) ∈ 𝑦))
25	24	expd 415	. . . . . . . . . . . . . . 15 ⊢ (Tr 𝑦 → ((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) → ((𝐹‘𝑧) ∈ 𝑦 → (𝐹‘suc 𝑧) ∈ 𝑦)))
26	25	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) → (Tr 𝑦 → ((𝐹‘𝑧) ∈ 𝑦 → (𝐹‘suc 𝑧) ∈ 𝑦)))
27	23, 26	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (Tr 𝑦 → ((𝐹‘𝑧) ∈ 𝑦 → (𝐹‘suc 𝑧) ∈ 𝑦))))
28	27	impd 410	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ Tr 𝑦) → ((𝐹‘𝑧) ∈ 𝑦 → (𝐹‘suc 𝑧) ∈ 𝑦)))
29	18, 28	syl5 34	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ((𝐹‘𝑧) ∈ 𝑦 → (𝐹‘suc 𝑧) ∈ 𝑦)))
30	6, 8, 10, 17, 29	finds2 7843	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹‘𝑤) ∈ 𝑦))
31	30	com12 32	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ ω → (𝐹‘𝑤) ∈ 𝑦))
32	31	ralrimiv 3129	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦)
33	32	3expib 1123	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
34	33	adantl 481	. . . . . 6 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
35		f1fun 6733	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → Fun 𝐹)
36		f1dm 6735	. . . . . . . . 9 ⊢ (𝐹:ω–1-1→V → dom 𝐹 = ω)
37	36	eqimsscd 3980	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → ω ⊆ dom 𝐹)
38		funimass4 6899	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ω ⊆ dom 𝐹) → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
39	35, 37, 38	syl2anc 585	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
40	39	adantr 480	. . . . . 6 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
41	34, 40	sylibrd 259	. . . . 5 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹 “ ω) ⊆ 𝑦))
42		ominf 9168	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
43		f1fn 6732	. . . . . . . . . . . . . 14 ⊢ (𝐹:ω–1-1→V → 𝐹 Fn ω)
44		fnima 6623	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn ω → (𝐹 “ ω) = ran 𝐹)
45	43, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐹:ω–1-1→V → (𝐹 “ ω) = ran 𝐹)
46	45	eqimsscd 3980	. . . . . . . . . . . 12 ⊢ (𝐹:ω–1-1→V → ran 𝐹 ⊆ (𝐹 “ ω))
47		f1ssr 6737	. . . . . . . . . . . 12 ⊢ ((𝐹:ω–1-1→V ∧ ran 𝐹 ⊆ (𝐹 “ ω)) → 𝐹:ω–1-1→(𝐹 “ ω))
48	46, 47	mpdan 688	. . . . . . . . . . 11 ⊢ (𝐹:ω–1-1→V → 𝐹:ω–1-1→(𝐹 “ ω))
49		f1fi 9218	. . . . . . . . . . 11 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→(𝐹 “ ω)) → ω ∈ Fin)
50	48, 49	sylan2 594	. . . . . . . . . 10 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→V) → ω ∈ Fin)
51	50	ancoms 458	. . . . . . . . 9 ⊢ ((𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin) → ω ∈ Fin)
52	42, 51	mto 197	. . . . . . . 8 ⊢ ¬ (𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin)
53	52	imnani 400	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ¬ (𝐹 “ ω) ∈ Fin)
54		ssfi 9101	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ (𝐹 “ ω) ⊆ 𝑦) → (𝐹 “ ω) ∈ Fin)
55	54	ancoms 458	. . . . . . . . 9 ⊢ (((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin) → (𝐹 “ ω) ∈ Fin)
56	55	con3i 154	. . . . . . . 8 ⊢ (¬ (𝐹 “ ω) ∈ Fin → ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
57		imnan 399	. . . . . . . 8 ⊢ (((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin) ↔ ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
58	56, 57	sylibr 234	. . . . . . 7 ⊢ (¬ (𝐹 “ ω) ∈ Fin → ((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin))
59	53, 58	syl 17	. . . . . 6 ⊢ (𝐹:ω–1-1→V → ((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin))
60	59	adantr 480	. . . . 5 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin))
61	41, 60	syld 47	. . . 4 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ¬ 𝑦 ∈ Fin))
62	61	3adant1 1131	. . 3 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ¬ 𝑦 ∈ Fin))
63	4, 62	mt2d 136	. 2 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
64	63	nexdv 1938	1 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  {csn 4568  Tr wtr 5193  dom cdm 5625  ran crn 5626   “ cima 5628  suc csuc 6320  Fun wfun 6487   Fn wfn 6488  –1-1→wf1 6490  ‘cfv 6493  ωcom 7811  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7812  df-1o 8399  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891

This theorem is referenced by: (None)