Theorem allbutfi 45579

Description: For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 45285 and eliuniin2 45306 (here, the precondition can be dropped; see eliuniincex 45295). (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
allbutfi.z	⊢ 𝑍 = (ℤ≥‘𝑀)
allbutfi.a	⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵

Assertion

Ref	Expression
allbutfi	⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)

Distinct variable group:   𝑚,𝑋,𝑛

Allowed substitution hints:   𝐴(𝑚,𝑛)   𝐵(𝑚,𝑛)   𝑀(𝑚,𝑛)   𝑍(𝑚,𝑛)

Proof of Theorem allbutfi

Step	Hyp	Ref	Expression
1		allbutfi.a	. . . . . 6 ⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵
2	1	eleq2i 2826	. . . . 5 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
3	2	biimpi 216	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
4		eliun 4948	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
5	3, 4	sylib 218	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
6		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑛𝑋
7		nfiu1 4980	. . . . . 6 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵
8	1, 7	nfcxfr 2894	. . . . 5 ⊢ Ⅎ𝑛𝐴
9	6, 8	nfel 2911	. . . 4 ⊢ Ⅎ𝑛 𝑋 ∈ 𝐴
10		eliin 4949	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
11	10	biimpd 229	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
12	11	a1d 25	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)))
13	9, 12	reximdai 3236	. . 3 ⊢ (𝑋 ∈ 𝐴 → (∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
14	5, 13	mpd 15	. 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)
15		simpr 484	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)
16		allbutfi.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
17	16	eleq2i 2826	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19		eluzelz 12759	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
20		uzid 12764	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	18, 19, 20	3syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	ne0d 4292	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
23		eliin2 45302	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑛) ≠ ∅ → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
25	24	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
26	15, 25	mpbird 257	. . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
27	26	ex 412	. . . . 5 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵))
28	27	reximia 3069	. . . 4 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
29	28, 4	sylibr 234	. . 3 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
30	29, 1	eleqtrrdi 2845	. 2 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐴)
31	14, 30	impbii 209	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  ∅c0 4283  ∪ ciun 4944  ∩ ciin 4945  ‘cfv 6490  ℤcz 12486  ℤ_≥cuz 12749

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-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-pre-lttri 11098

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-neg 11365  df-z 12487  df-uz 12750

This theorem is referenced by:  allbutfiinf  45606  allbutfifvre  45861  smflimlem3  46959  smfliminflem  47016