Theorem noinfepregs 35289

Description: There are no infinite descending ∈-chains, proven using ax-regs 35282. (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 35288	. . 3 ⊢ ((𝐹 ↾ ω) Fn ω → ∃𝑥 ∈ ω ((𝐹 ↾ ω)‘suc 𝑥) ∉ ((𝐹 ↾ ω)‘𝑥))
2		peano2 7832	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
3	2	fvresd 6854	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝐹 ↾ ω)‘suc 𝑥) = (𝐹‘suc 𝑥))
4		fvres 6853	. . . . 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 3081	. . 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 6619	. . . . 5 ⊢ ((𝐹 ↾ ω) Fn ω ↔ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
9	8	notbii 320	. . . 4 ⊢ (¬ (𝐹 ↾ ω) Fn ω ↔ ¬ ∀𝑥 ∈ ω ∃!𝑦 𝑥𝐹𝑦)
10		rexnal 3088	. . . 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 6821	. . . . 5 ⊢ (¬ ∃!𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = ∅)
13		nel02 4291	. . . . . 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 3074	. . 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 1541   ∈ wcel 2113  ∃!weu 2568   ∉ wnel 3036  ∀wral 3051  ∃wrex 3060  ∅c0 4285   class class class wbr 5098   ↾ cres 5626  suc csuc 6319   Fn wfn 6487  ‘cfv 6492  ωcom 7808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-regs 35282

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-om 7809

This theorem is referenced by: (None)