Theorem noinfepfnregs 35288

Description: There are no infinite descending ∈-chains, proven using ax-regs 35282. (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 7831	. . . . . 6 ⊢ ∅ ∈ ω
2	1	n0ii 4295	. . . . 5 ⊢ ¬ ω = ∅
3		ssid 3956	. . . . . 6 ⊢ ω ⊆ ω
4		fnimaeq0 6625	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
5	3, 4	mpan2 691	. . . . 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 35285	. . 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 6906	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
11	3, 10	mpan2 691	. . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → 𝑥 ∈ ω)
14		peano2 7832	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
15		fnfvima 7179	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
16	3, 15	mp3an2 1451	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
17	14, 16	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ω ∧ 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
18	17	ad2ant2r 747	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
19		ineq1 4165	. . . . . . . . . . . . . 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 746	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
23		minel 4418	. . . . . . . . . . 11 ⊢ (((𝐹‘suc 𝑥) ∈ (𝐹 “ ω) ∧ ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
24	18, 22, 23	syl2anc 584	. . . . . . . . . 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 3146	. . . . . 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 3143	1 ⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∉ wnel 3036  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285   “ cima 5627  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:  noinfepregs  35289