Theorem nninfnfiinf 17066

Description: An element of ℕ_∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)

Assertion

Ref	Expression
nninfnfiinf	⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o))

Distinct variable group:   𝐴,𝑖,𝑛

Proof of Theorem nninfnfiinf

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 533	. . . . . 6 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
2		simplll 539	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → 𝐴 ∈ ℕ∞)
3		simplr 533	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → 𝑗 ∈ ω)
4		simpr 110	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → (𝐴‘𝑗) = ∅)
5	2, 3, 4	nnnninfex 17065	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
6	1, 5	mtand 675	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ¬ (𝐴‘𝑗) = ∅)
7		nninff 7462	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o)
8	7	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → 𝐴:ω⟶2o)
9		simpr 110	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
10	8, 9	ffvelcdmd 5844	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ 2o)
11		df2o3 6702	. . . . . . . 8 ⊢ 2o = {∅, 1o}
12	10, 11	eleqtrdi 2331	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ {∅, 1o})
13		elpri 3732	. . . . . . 7 ⊢ ((𝐴‘𝑗) ∈ {∅, 1o} → ((𝐴‘𝑗) = ∅ ∨ (𝐴‘𝑗) = 1o))
14	12, 13	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = ∅ ∨ (𝐴‘𝑗) = 1o))
15	14	orcomd 741	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = 1o ∨ (𝐴‘𝑗) = ∅))
16	6, 15	ecased 1390	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = 1o)
17		fconstmpt 4822	. . . . . . 7 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
18	17	fveq1i 5696	. . . . . 6 ⊢ ((ω × {1o})‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)
19		1oex 6695	. . . . . . 7 ⊢ 1o ∈ V
20	19	fvconst2 5931	. . . . . 6 ⊢ (𝑗 ∈ ω → ((ω × {1o})‘𝑗) = 1o)
21	18, 20	eqtr3id 2285	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
22	21	adantl 277	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
23	16, 22	eqtr4d 2274	. . 3 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	23	ralrimiva 2623	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
25	7	ffnd 5534	. . . 4 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 Fn ω)
26		eqid 2238	. . . . 5 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
27	19, 26	fnmpti 5512	. . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
28		eqfnfv 5806	. . . 4 ⊢ ((𝐴 Fn ω ∧ (𝑖 ∈ ω ↦ 1o) Fn ω) → (𝐴 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
29	25, 27, 28	sylancl 417	. . 3 ⊢ (𝐴 ∈ ℕ∞ → (𝐴 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
30	29	adantr 276	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → (𝐴 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
31	24, 30	mpbird 167	1 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∅c0 3520  ifcif 3638  {csn 3709  {cpr 3710   ↦ cmpt 4192  ωcom 4737   × cxp 4772   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681  ℕ_∞xnninf 7459

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1o 6687  df-2o 6688  df-map 6924  df-nninf 7460

This theorem is used by: (None)