Theorem allbutfi 45851

Description: For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 45560 and eliuniin2 45581 (here, the precondition can be dropped; see eliuniincex 45570). (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 2833	. . . . 5 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
3	2	biimpi 218	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
4		eliun 4928	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
5	3, 4	sylib 220	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
6		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑛𝑋
7		nfiu1 4960	. . . . . 6 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵
8	1, 7	nfcxfr 2901	. . . . 5 ⊢ Ⅎ𝑛𝐴
9	6, 8	nfel 2917	. . . 4 ⊢ Ⅎ𝑛 𝑋 ∈ 𝐴
10		eliin 4929	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
11	10	biimpd 231	. . . . 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 3243	. . 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 486	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)
16		allbutfi.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
17	16	eleq2i 2833	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19		eluzelz 12793	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
20		uzid 12798	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	18, 19, 20	3syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	ne0d 4273	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
23		eliin2 45577	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
26	15, 25	mpbird 259	. . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
27	26	ex 414	. . . . 5 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵))
28	27	reximia 3076	. . . 4 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
29	28, 4	sylibr 236	. . 3 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
30	29, 1	eleqtrrdi 2852	. 2 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐴)
31	14, 30	impbii 211	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  ∅c0 4264  ∪ ciun 4924  ∩ ciin 4925  ‘cfv 6489  ℤcz 12519  ℤ_≥cuz 12783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-pre-lttri 11107

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-neg 11375  df-z 12520  df-uz 12784

This theorem is referenced by:  allbutfiinf  45877  allbutfifvre  46132  smflimlem3  47230  smfliminflem  47287