Theorem noinfepregs 35277

Description: There are no infinite descending ∈-chains, proven using ax-regs 35270. (Contributed by BTernaryTau, 18-Feb-2026.)

Assertion

Ref	Expression
noinfepregs	⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)

Distinct variable group:   𝑥,𝐹

Proof of Theorem noinfepregs

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noinfepfnregs 35276	. . 3 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥))
2		peano2 7841	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
3	2	fvresd 6860	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘suc 𝑥) = (𝐹‘suc 𝑥))
4		fvres 6859	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘𝑥) = (𝐹‘𝑥))
5	3, 4	neleq12d 3041	. . . 4 ⊢ (𝑥 ∈ ω → (((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥) ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
6	5	rexbiia 3082	. . 3 ⊢ (∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥) ↔ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
7	1, 6	sylib 218	. 2 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
8		fnres 6625	. . . . 5 ⊢ ((𝐹 ↾ ω) Fn ω ↔ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
9	8	notbii 320	. . . 4 ⊢ (¬ (𝐹 ↾ ω) Fn ω ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
10		rexnal 3089	. . . 4 ⊢ (∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦 ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
11	9, 10	sylbb2 238	. . 3 ⊢ (¬ (𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦)
12		tz6.12-2 6827	. . . . 5 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = ∅)
13		nel02 4279	. . . . . 6 ⊢ ((𝐹‘𝑥) = ∅ → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
14		df-nel 3037	. . . . . 6 ⊢ ((𝐹‘suc 𝑥) ∉ (𝐹‘𝑥) ↔ ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
15	13, 14	sylibr 234	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
16	12, 15	syl 17	. . . 4 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
17	16	reximi 3075	. . 3 ⊢ (∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
18	11, 17	syl 17	. 2 ⊢ (¬ (𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
19	7, 18	pm2.61i 182	1 ⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∃!weu 2568   ∉ wnel 3036  ∀wral 3051  ∃wrex 3061  ∅c0 4273   class class class wbr 5085   ↾ cres 5633  suc csuc 6325   Fn wfn 6493  ‘cfv 6498  ωcom 7817

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-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-regs 35270

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-om 7818

This theorem is referenced by: (None)