Theorem noinfepregs 35546

Description: There are no infinite descending ∈-chains, proven using ax-regs 35539. (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 35545	. . 3 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥))
2		peano2 7882	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
3	2	fvresd 6901	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘suc 𝑥) = (𝐹‘suc 𝑥))
4		fvres 6900	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘𝑥) = (𝐹‘𝑥))
5	3, 4	neleq12d 3069	. . . 4 ⊢ (𝑥 ∈ ω → (((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥) ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
6	5	rexbiia 3110	. . 3 ⊢ (∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥) ↔ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
7	1, 6	sylib 221	. 2 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
8		fnres 6662	. . . . 5 ⊢ ((𝐹 ↾ ω) Fn ω ↔ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
9	8	notbii 323	. . . 4 ⊢ (¬ (𝐹 ↾ ω) Fn ω ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
10		rexnal 3117	. . . 4 ⊢ (∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦 ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
11	9, 10	sylbb2 241	. . 3 ⊢ (¬ (𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦)
12		tz6.12-2 6868	. . . . 5 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = ∅)
13		nel02 4292	. . . . . 6 ⊢ ((𝐹‘𝑥) = ∅ → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
14		df-nel 3065	. . . . . 6 ⊢ ((𝐹‘suc 𝑥) ∉ (𝐹‘𝑥) ↔ ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
15	13, 14	sylibr 237	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
16	12, 15	syl 18	. . . 4 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
17	16	reximi 3103	. . 3 ⊢ (∃𝑥 ∈ ω ¬ ∃!𝑦 𝑥𝐹𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
18	11, 17	syl 18	. 2 ⊢ (¬ (𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
19	7, 18	pm2.61i 184	1 ⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  ∃!weu 2596   ∉ wnel 3064  ∀wral 3079  ∃wrex 3089  ∅c0 4286   class class class wbr 5109   ↾ cres 5663  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732  ax-regs 35539

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7859

This theorem is referenced by: (None)