Theorem allbutfi 41672

Description: For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 41372 and eliuniin2 41393 (here, the precondition can be dropped; see eliuniincex 41382). (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 2906	. . . . 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 4925	. . . 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 2979	. . . . 5 ⊢ Ⅎ𝑛𝑋
7		nfiu1 4955	. . . . . 6 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵
8	1, 7	nfcxfr 2977	. . . . 5 ⊢ Ⅎ𝑛𝐴
9	6, 8	nfel 2994	. . . 4 ⊢ Ⅎ𝑛 𝑋 ∈ 𝐴
10		eliin 4926	. . . . . 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 3313	. . 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 487	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)
16		allbutfi.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
17	16	eleq2i 2906	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19		eluzelz 12256	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
20		uzid 12261	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	18, 19, 20	3syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	ne0d 4303	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
23		eliin2 41389	. . . . . . . . 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 483	. . . . . . 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 415	. . . . 5 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵))
28	27	reximia 3244	. . . 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 2926	. 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 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  ∅c0 4293  ∪ ciun 4921  ∩ ciin 4922  ‘cfv 6357  ℤcz 11984  ℤ_≥cuz 12246

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-pre-lttri 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-neg 10875  df-z 11985  df-uz 12247

This theorem is referenced by:  allbutfiinf  41701  allbutfifvre  41963  smflimlem3  43056  smfliminflem  43111