Theorem allbutfi 44038

Description: For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 43721 and eliuniin2 43742 (here, the precondition can be dropped; see eliuniincex 43731). (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 215	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
4		eliun 5000	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
5	3, 4	sylib 217	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
6		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑛𝑋
7		nfiu1 5030	. . . . . 6 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵
8	1, 7	nfcxfr 2902	. . . . 5 ⊢ Ⅎ𝑛𝐴
9	6, 8	nfel 2918	. . . 4 ⊢ Ⅎ𝑛 𝑋 ∈ 𝐴
10		eliin 5001	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))
11	10	biimpd 228	. . . . 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 3259	. . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19		eluzelz 12828	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
20		uzid 12833	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	18, 19, 20	3syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	ne0d 4334	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
23		eliin2 43738	. . . . . . . . 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 257	. . . . . 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 3082	. . . 4 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)
29	28, 4	sylibr 233	. . 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 208	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  ∅c0 4321  ∪ ciun 4996  ∩ ciin 4997  ‘cfv 6540  ℤcz 12554  ℤ_≥cuz 12818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-pre-lttri 11180

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-neg 11443  df-z 12555  df-uz 12819

This theorem is referenced by:  allbutfiinf  44065  allbutfifvre  44326  smflimlem3  45424  smfliminflem  45481