Theorem noinfepregs 35296

Description: There are no infinite descending ∈-chains, proven using ax-regs 35289. (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 35295	. . 3 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥))
2		peano2 7835	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
3	2	fvresd 6855	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘suc 𝑥) = (𝐹‘suc 𝑥))
4		fvres 6854	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘𝑥) = (𝐹‘𝑥))
5	3, 4	neleq12d 3042	. . . 4 ⊢ (𝑥 ∈ ω → (((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥) ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
6	5	rexbiia 3083	. . 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 6620	. . . . 5 ⊢ ((𝐹 ↾ ω) Fn ω ↔ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
9	8	notbii 320	. . . 4 ⊢ (¬ (𝐹 ↾ ω) Fn ω ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
10		rexnal 3090	. . . 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 6822	. . . . 5 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = ∅)
13		nel02 4280	. . . . . 6 ⊢ ((𝐹‘𝑥) = ∅ → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
14		df-nel 3038	. . . . . 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 3076	. . 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 2569   ∉ wnel 3037  ∀wral 3052  ∃wrex 3062  ∅c0 4274   class class class wbr 5086   ↾ cres 5627  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  ωcom 7811

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 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683  ax-regs 35289

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-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  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-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-fv 6501  df-om 7812

This theorem is referenced by: (None)