Theorem nninfnfiinf 16927

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 529	. . . . . 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 535	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → 𝐴 ∈ ℕ∞)
3		simplr 529	. . . . . . 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 16926	. . . . . 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 671	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ¬ (𝐴‘𝑗) = ∅)
7		nninff 7426	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o)
8	7	ad2antrr 488	. . . . . . . . 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 5818	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ 2o)
11		df2o3 6675	. . . . . . . 8 ⊢ 2o = {∅, 1o}
12	10, 11	eleqtrdi 2327	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ {∅, 1o})
13		elpri 3717	. . . . . . 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 737	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = 1o ∨ (𝐴‘𝑗) = ∅))
16	6, 15	ecased 1386	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = 1o)
17		fconstmpt 4802	. . . . . . 7 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
18	17	fveq1i 5676	. . . . . 6 ⊢ ((ω × {1o})‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)
19		1oex 6668	. . . . . . 7 ⊢ 1o ∈ V
20	19	fvconst2 5905	. . . . . 6 ⊢ (𝑗 ∈ ω → ((ω × {1o})‘𝑗) = 1o)
21	18, 20	eqtr3id 2281	. . . . 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 2270	. . 3 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	23	ralrimiva 2617	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
25	7	ffnd 5514	. . . 4 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 Fn ω)
26		eqid 2234	. . . . 5 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
27	19, 26	fnmpti 5492	. . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
28		eqfnfv 5780	. . . 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 413	. . 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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ∅c0 3512  ifcif 3624  {csn 3694  {cpr 3695   ↦ cmpt 4176  ωcom 4717   × cxp 4752   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654  ℕ_∞xnninf 7423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1o 6660  df-2o 6661  df-map 6897  df-nninf 7424

This theorem is referenced by: (None)