Theorem fineqvinfep 35306

Description: A counterexample demonstrating that tz9.1 9641 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 3435	. . . . 5 ⊢ 𝑦 ∈ V
2		eleq2 2828	. . . . 5 ⊢ (Fin = V → (𝑦 ∈ Fin ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 259	. . . 4 ⊢ (Fin = V → 𝑦 ∈ Fin)
4	3	3ad2ant1 1139	. . 3 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → 𝑦 ∈ Fin)
5		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝐹‘𝑤) = (𝐹‘∅))
6	5	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘∅) ∈ 𝑦))
7		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧))
8	7	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘𝑧) ∈ 𝑦))
9		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → (𝐹‘𝑤) = (𝐹‘suc 𝑧))
10	9	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘suc 𝑧) ∈ 𝑦))
11		simp2 1143	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝐴 ⊆ 𝑦)
12		fvex 6840	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘∅) ∈ V
13	12	snid 4594	. . . . . . . . . . . . . 14 ⊢ (𝐹‘∅) ∈ {(𝐹‘∅)}
14		fineqvinfep.1	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {(𝐹‘∅)}
15	13, 14	eleqtrri 2838	. . . . . . . . . . . . 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 3916	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹‘∅) ∈ 𝑦)
18		3simpb 1155	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ Tr 𝑦))
19		suceq 6378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
20	19	fveq2d 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
21		fveq2 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
22	20, 21	eleq12d 2833	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
23	22	rspcv 3556	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
24		trel 5187	. . . . . . . . . . . . . . . 16 ⊢ (Tr 𝑦 → (((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ∈ 𝑦) → (𝐹‘suc 𝑧) ∈ 𝑦))
25	24	expd 416	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 7838	. . . . . . . . . 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 3130	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦)
33	32	3expib 1128	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
34	33	adantl 482	. . . . . 6 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
35		f1fun 6725	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → Fun 𝐹)
36		f1dm 6727	. . . . . . . . 9 ⊢ (𝐹:ω–1-1→V → dom 𝐹 = ω)
37	36	eqimsscd 3972	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → ω ⊆ dom 𝐹)
38		funimass4 6891	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ω ⊆ dom 𝐹) → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
39	35, 37, 38	syl2anc 590	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
40	39	adantr 481	. . . . . 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 260	. . . . 5 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹 “ ω) ⊆ 𝑦))
42		ominf 9164	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
43		f1fn 6724	. . . . . . . . . . . . . 14 ⊢ (𝐹:ω–1-1→V → 𝐹 Fn ω)
44		fnima 6615	. . . . . . . . . . . . . 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 3972	. . . . . . . . . . . 12 ⊢ (𝐹:ω–1-1→V → ran 𝐹 ⊆ (𝐹 “ ω))
47		f1ssr 6729	. . . . . . . . . . . 12 ⊢ ((𝐹:ω–1-1→V ∧ ran 𝐹 ⊆ (𝐹 “ ω)) → 𝐹:ω–1-1→(𝐹 “ ω))
48	46, 47	mpdan 693	. . . . . . . . . . 11 ⊢ (𝐹:ω–1-1→V → 𝐹:ω–1-1→(𝐹 “ ω))
49		f1fi 9214	. . . . . . . . . . 11 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→(𝐹 “ ω)) → ω ∈ Fin)
50	48, 49	sylan2 599	. . . . . . . . . 10 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→V) → ω ∈ Fin)
51	50	ancoms 459	. . . . . . . . 9 ⊢ ((𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin) → ω ∈ Fin)
52	42, 51	mto 198	. . . . . . . 8 ⊢ ¬ (𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin)
53	52	imnani 401	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ¬ (𝐹 “ ω) ∈ Fin)
54		ssfi 9097	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ (𝐹 “ ω) ⊆ 𝑦) → (𝐹 “ ω) ∈ Fin)
55	54	ancoms 459	. . . . . . . . 9 ⊢ (((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin) → (𝐹 “ ω) ∈ Fin)
56	55	con3i 154	. . . . . . . 8 ⊢ (¬ (𝐹 “ ω) ∈ Fin → ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
57		imnan 400	. . . . . . . 8 ⊢ (((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin) ↔ ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
58	56, 57	sylibr 235	. . . . . . 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 481	. . . . 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 1136	. . 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 1943	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  {csn 4555  Tr wtr 5179  dom cdm 5618  ran crn 5619   “ cima 5621  suc csuc 6312  Fun wfun 6479   Fn wfn 6480  –1-1→wf1 6482  ‘cfv 6485  ωcom 7806  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by: (None)