Theorem noinfepfnregs 35276

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

Assertion

Ref	Expression
noinfepfnregs	⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

Distinct variable group:   𝑥,𝐹

Proof of Theorem noinfepfnregs

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7840	. . . . . 6 ⊢ ∅ ∈ ω
2	1	n0ii 4283	. . . . 5 ⊢ ¬ ω = ∅
3		ssid 3944	. . . . . 6 ⊢ ω ⊆ ω
4		fnimaeq0 6631	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
5	3, 4	mpan2 692	. . . . 5 ⊢ (𝐹 Fn ω → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
6	2, 5	mtbiri 327	. . . 4 ⊢ (𝐹 Fn ω → ¬ (𝐹 “ ω) = ∅)
7	6	neqned 2939	. . 3 ⊢ (𝐹 Fn ω → (𝐹 “ ω) ≠ ∅)
8		axregszf 35273	. . 3 ⊢ ((𝐹 “ ω) ≠ ∅ → ∃𝑦 ∈ (𝐹 “ ω)(𝑦 ∩ (𝐹 “ ω)) = ∅)
9	7, 8	syl 17	. 2 ⊢ (𝐹 Fn ω → ∃𝑦 ∈ (𝐹 “ ω)(𝑦 ∩ (𝐹 “ ω)) = ∅)
10		fvelimab 6912	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
11	3, 10	mpan2 692	. . . . . . 7 ⊢ (𝐹 Fn ω → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
12	11	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
13		simprl 771	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → 𝑥 ∈ ω)
14		peano2 7841	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
15		fnfvima 7188	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
16	3, 15	mp3an2 1452	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
17	14, 16	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ω ∧ 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
18	17	ad2ant2r 748	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
19		ineq1 4153	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = (𝑦 ∩ (𝐹 “ ω)))
20	19	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅ ↔ (𝑦 ∩ (𝐹 “ ω)) = ∅))
21	20	biimparc 479	. . . . . . . . . . . 12 ⊢ (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
22	21	ad2ant2l 747	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
23		minel 4406	. . . . . . . . . . 11 ⊢ (((𝐹‘suc 𝑥) ∈ (𝐹 “ ω) ∧ ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
24	18, 22, 23	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
25		df-nel 3037	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ (𝐹‘𝑥) ↔ ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
26	24, 25	sylibr 234	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
27	13, 26	jca 511	. . . . . . . 8 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
28	27	ex 412	. . . . . . 7 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ((𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
29	28	reximdv2 3147	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
30	12, 29	sylbid 240	. . . . 5 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (𝑦 ∈ (𝐹 “ ω) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
31	30	expimpd 453	. . . 4 ⊢ (𝐹 Fn ω → (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ 𝑦 ∈ (𝐹 “ ω)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
32	31	ancomsd 465	. . 3 ⊢ (𝐹 Fn ω → ((𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
33	32	imp 406	. 2 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
34	9, 33	rexlimddv 3144	1 ⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ∉ wnel 3036  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   “ cima 5634  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:  noinfepregs  35277