Theorem fineqvinfep 35421

Description: A counterexample demonstrating that tz9.1 9684 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 3458	. . . . 5 ⊢ 𝑦 ∈ V
2		eleq2 2851	. . . . 5 ⊢ (Fin = V → (𝑦 ∈ Fin ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 260	. . . 4 ⊢ (Fin = V → 𝑦 ∈ Fin)
4	3	3ad2ant1 1146	. . 3 ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → 𝑦 ∈ Fin)
5		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝐹‘𝑤) = (𝐹‘∅))
6	5	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘∅) ∈ 𝑦))
7		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧))
8	7	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘𝑧) ∈ 𝑦))
9		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → (𝐹‘𝑤) = (𝐹‘suc 𝑧))
10	9	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑧 → ((𝐹‘𝑤) ∈ 𝑦 ↔ (𝐹‘suc 𝑧) ∈ 𝑦))
11		simp2 1150	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝐴 ⊆ 𝑦)
12		fvex 6880	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘∅) ∈ V
13	12	snid 4621	. . . . . . . . . . . . . 14 ⊢ (𝐹‘∅) ∈ {(𝐹‘∅)}
14		fineqvinfep.1	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {(𝐹‘∅)}
15	13, 14	eleqtrri 2861	. . . . . . . . . . . . 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 3937	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹‘∅) ∈ 𝑦)
18		3simpb 1162	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ Tr 𝑦))
19		suceq 6414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
20	19	fveq2d 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
21		fveq2 6867	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
22	20, 21	eleq12d 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
23	22	rspcv 3577	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
24		trel 5215	. . . . . . . . . . . . . . . 16 ⊢ (Tr 𝑦 → (((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ∈ 𝑦) → (𝐹‘suc 𝑧) ∈ 𝑦))
25	24	expd 419	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 7879	. . . . . . . . . 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 3153	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦)
33	32	3expib 1135	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
34	33	adantl 485	. . . . . 6 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
35		f1fun 6762	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → Fun 𝐹)
36		f1dm 6766	. . . . . . . . 9 ⊢ (𝐹:ω–1-1→V → dom 𝐹 = ω)
37	36	eqimsscd 3993	. . . . . . . 8 ⊢ (𝐹:ω–1-1→V → ω ⊆ dom 𝐹)
38		funimass4 6931	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ω ⊆ dom 𝐹) → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
39	35, 37, 38	syl2anc 593	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ((𝐹 “ ω) ⊆ 𝑦 ↔ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ 𝑦))
40	39	adantr 484	. . . . . 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 261	. . . . 5 ⊢ ((𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → (𝐹 “ ω) ⊆ 𝑦))
42		ominf 9208	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
43		f1fn 6761	. . . . . . . . . . . . . 14 ⊢ (𝐹:ω–1-1→V → 𝐹 Fn ω)
44		fnima 6651	. . . . . . . . . . . . . 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 3993	. . . . . . . . . . . 12 ⊢ (𝐹:ω–1-1→V → ran 𝐹 ⊆ (𝐹 “ ω))
47		f1ssr 6768	. . . . . . . . . . . 12 ⊢ ((𝐹:ω–1-1→V ∧ ran 𝐹 ⊆ (𝐹 “ ω)) → 𝐹:ω–1-1→(𝐹 “ ω))
48	46, 47	mpdan 697	. . . . . . . . . . 11 ⊢ (𝐹:ω–1-1→V → 𝐹:ω–1-1→(𝐹 “ ω))
49		f1fi 9258	. . . . . . . . . . 11 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→(𝐹 “ ω)) → ω ∈ Fin)
50	48, 49	sylan2 602	. . . . . . . . . 10 ⊢ (((𝐹 “ ω) ∈ Fin ∧ 𝐹:ω–1-1→V) → ω ∈ Fin)
51	50	ancoms 462	. . . . . . . . 9 ⊢ ((𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin) → ω ∈ Fin)
52	42, 51	mto 199	. . . . . . . 8 ⊢ ¬ (𝐹:ω–1-1→V ∧ (𝐹 “ ω) ∈ Fin)
53	52	imnani 404	. . . . . . 7 ⊢ (𝐹:ω–1-1→V → ¬ (𝐹 “ ω) ∈ Fin)
54		ssfi 9141	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ (𝐹 “ ω) ⊆ 𝑦) → (𝐹 “ ω) ∈ Fin)
55	54	ancoms 462	. . . . . . . . 9 ⊢ (((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin) → (𝐹 “ ω) ∈ Fin)
56	55	con3i 154	. . . . . . . 8 ⊢ (¬ (𝐹 “ ω) ∈ Fin → ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
57		imnan 403	. . . . . . . 8 ⊢ (((𝐹 “ ω) ⊆ 𝑦 → ¬ 𝑦 ∈ Fin) ↔ ¬ ((𝐹 “ ω) ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
58	56, 57	sylibr 236	. . . . . . 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 484	. . . . 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 1143	. . 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 1956	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 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  {csn 4582  Tr wtr 5207  dom cdm 5647  ran crn 5648   “ cima 5650  suc csuc 6348  Fun wfun 6515   Fn wfn 6516  –1-1→wf1 6518  ‘cfv 6521  ωcom 7846  Fincfn 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  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-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-1o 8437  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931

This theorem is referenced by: (None)