Theorem noinfepfnregs 35800

Description: There are no infinite descending ∈-chains, proven using ax-regs 35794. (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 7900	. . . . . 6 ⊢ ∅ ∈ ω
2	1	n0ii 4289	. . . . 5 ⊢ ¬ ω = ∅
3		ssid 3953	. . . . . 6 ⊢ ω ⊆ ω
4		fnimaeq0 6672	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
5	3, 4	mpan2 704	. . . . 5 ⊢ (𝐹 Fn ω → ((𝐹 “ ω) = ∅ ↔ ω = ∅))
6	2, 5	mtbiri 330	. . . 4 ⊢ (𝐹 Fn ω → ¬ (𝐹 “ ω) = ∅)
7	6	neqned 2963	. . 3 ⊢ (𝐹 Fn ω → (𝐹 “ ω) ≠ ∅)
8		axregszf 35797	. . 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 6957	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
11	3, 10	mpan2 704	. . . . . . 7 ⊢ (𝐹 Fn ω → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
12	11	adantr 486	. . . . . 6 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → (𝑦 ∈ (𝐹 “ ω) ↔ ∃𝑥 ∈ ω (𝐹‘𝑥) = 𝑦))
13		simprl 783	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → 𝑥 ∈ ω)
14		peano2 7901	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
15		fnfvima 7239	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn ω ∧ ω ⊆ ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
16	3, 15	mp3an2 1478	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn ω ∧ suc 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
17	14, 16	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ω ∧ 𝑥 ∈ ω) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
18	17	ad2ant2r 760	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝐹‘suc 𝑥) ∈ (𝐹 “ ω))
19		ineq1 4159	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = (𝑦 ∩ (𝐹 “ ω)))
20	19	eqeq1d 2763	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅ ↔ (𝑦 ∩ (𝐹 “ ω)) = ∅))
21	20	biimparc 485	. . . . . . . . . . . 12 ⊢ (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
22	21	ad2ant2l 759	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅)
23		minel 4419	. . . . . . . . . . 11 ⊢ (((𝐹‘suc 𝑥) ∈ (𝐹 “ ω) ∧ ((𝐹‘𝑥) ∩ (𝐹 “ ω)) = ∅) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
24	18, 22, 23	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
25		df-nel 3063	. . . . . . . . . 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 521	. . . . . . . 8 ⊢ (((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) ∧ (𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦)) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
28	27	ex 418	. . . . . . 7 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ((𝑥 ∈ ω ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ ω ∧ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
29	28	reximdv2 3173	. . . . . 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 459	. . . 4 ⊢ (𝐹 Fn ω → (((𝑦 ∩ (𝐹 “ ω)) = ∅ ∧ 𝑦 ∈ (𝐹 “ ω)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
32	31	ancomsd 471	. . 3 ⊢ (𝐹 Fn ω → ((𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
33	32	imp 412	. 2 ⊢ ((𝐹 Fn ω ∧ (𝑦 ∈ (𝐹 “ ω) ∧ (𝑦 ∩ (𝐹 “ ω)) = ∅)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
34	9, 33	rexlimddv 3170	1 ⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∉ wnel 3062  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   “ cima 5654  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751  ax-regs 35794

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-om 7878

This theorem is used by:  noinfepregs  35801