Theorem nninfnfiinf 16162

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 528	. . . . . 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 533	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) ∧ (𝐴‘𝑗) = ∅) → 𝐴 ∈ ℕ∞)
3		simplr 528	. . . . . . 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 16161	. . . . . 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 667	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ¬ (𝐴‘𝑗) = ∅)
7		nninff 7250	. . . . . . . . . 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 5739	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ 2o)
11		df2o3 6539	. . . . . . . 8 ⊢ 2o = {∅, 1o}
12	10, 11	eleqtrdi 2300	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ {∅, 1o})
13		elpri 3666	. . . . . . 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 731	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = 1o ∨ (𝐴‘𝑗) = ∅))
16	6, 15	ecased 1362	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = 1o)
17		fconstmpt 4740	. . . . . . 7 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
18	17	fveq1i 5600	. . . . . 6 ⊢ ((ω × {1o})‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)
19		1oex 6533	. . . . . . 7 ⊢ 1o ∈ V
20	19	fvconst2 5823	. . . . . 6 ⊢ (𝑗 ∈ ω → ((ω × {1o})‘𝑗) = 1o)
21	18, 20	eqtr3id 2254	. . . . 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 2243	. . 3 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	23	ralrimiva 2581	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
25	7	ffnd 5446	. . . 4 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 Fn ω)
26		eqid 2207	. . . . 5 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
27	19, 26	fnmpti 5424	. . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
28		eqfnfv 5700	. . . 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 710   = wceq 1373   ∈ wcel 2178  ∀wral 2486  ∃wrex 2487  ∅c0 3468  ifcif 3579  {csn 3643  {cpr 3644   ↦ cmpt 4121  ωcom 4656   × cxp 4691   Fn wfn 5285  ⟶wf 5286  ‘cfv 5290  1_oc1o 6518  2_oc2o 6519  ℕ_∞xnninf 7247

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1o 6525  df-2o 6526  df-map 6760  df-nninf 7248

This theorem is referenced by: (None)