Theorem noinfepfnregs 35478

Description: There are no infinite descending ∈-chains, proven using ax-regs 35472. (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 7885	. . . . . 6 ⊢ ∅ ∈ ω
2	1	n0ii 4302	. . . . 5 ⊢ ¬ ω = ∅
3		ssid 3965	. . . . . 6 ⊢ ω ⊆ ω
4		fnimaeq0 6669	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
5	3, 4	mpan2 703	. . . . 5 ⊢ (𝐹 Fn ω → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
6	2, 5	mtbiri 330	. . . 4 ⊢ (𝐹 Fn ω → ¬ (𝐹 “ ω) = ∅)
7	6	neqned 2971	. . 3 ⊢ (𝐹 Fn ω → (𝐹 “ ω) ≠ ∅)
8		axregszf 35475	. . 3 ⊢ ((𝐹 “ ω) ≠ ∅ → ∃𝑦 ∈ (𝐹 “ ω)(𝑦 ∩ (𝐹 “ ω)) = ∅)
9	7, 8	syl 18	. 2 ⊢ (𝐹 Fn ω → ∃𝑦 ∈ (𝐹 “ ω)(𝑦 ∩ (𝐹 “ ω)) = ∅)
10		fvelimab 6954	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
11	3, 10	mpan2 703	. . . . . . 7 ⊢ (𝐹 Fn ω → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
12	11	adantr 485	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
13		simprl 782	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → 𝑥 ∈ ω)
14		peano2 7886	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
15		fnfvima 7232	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
16	3, 15	mp3an2 1475	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
17	14, 16	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ω ∧ 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
18	17	ad2ant2r 759	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
19		ineq1 4172	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = (𝑦 ∩ (𝐹 “ ω)))
20	19	eqeq1d 2771	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅ ↔ (𝑦 ∩ (𝐹 “ ω)) = ∅))
21	20	biimparc 484	. . . . . . . . . . . 12 ⊢ (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
22	21	ad2ant2l 758	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
23		minel 4430	. . . . . . . . . . 11 ⊢ (((𝐹‘suc 𝑥) ∈ (𝐹 “ ω) ∧ ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
24	18, 22, 23	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
25		df-nel 3071	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ (𝐹‘𝑥) ↔ ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
26	24, 25	sylibr 237	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
27	13, 26	jca 520	. . . . . . . 8 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
28	27	ex 417	. . . . . . 7 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ((𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
29	28	reximdv2 3181	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
30	12, 29	sylbid 243	. . . . 5 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (𝑦 ∈ (𝐹 “ ω) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
31	30	expimpd 458	. . . 4 ⊢ (𝐹 Fn ω → (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ 𝑦 ∈ (𝐹 “ ω)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
32	31	ancomsd 470	. . 3 ⊢ (𝐹 Fn ω → ((𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
33	32	imp 411	. 2 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
34	9, 33	rexlimddv 3178	1 ⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∃wrex 3095   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   “ cima 5665  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-regs 35472

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-om 7863

This theorem is referenced by:  noinfepregs  35479