Theorem nninfnfiinf 16389

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 16388	. . . . . 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 669	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ¬ (𝐴‘𝑗) = ∅)
7		nninff 7289	. . . . . . . . . 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 5771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ 2o)
11		df2o3 6576	. . . . . . . 8 ⊢ 2o = {∅, 1o}
12	10, 11	eleqtrdi 2322	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ {∅, 1o})
13		elpri 3689	. . . . . . 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 734	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = 1o ∨ (𝐴‘𝑗) = ∅))
16	6, 15	ecased 1383	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = 1o)
17		fconstmpt 4766	. . . . . . 7 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
18	17	fveq1i 5628	. . . . . 6 ⊢ ((ω × {1o})‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)
19		1oex 6570	. . . . . . 7 ⊢ 1o ∈ V
20	19	fvconst2 5855	. . . . . 6 ⊢ (𝑗 ∈ ω → ((ω × {1o})‘𝑗) = 1o)
21	18, 20	eqtr3id 2276	. . . . 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 2265	. . 3 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	23	ralrimiva 2603	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
25	7	ffnd 5474	. . . 4 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 Fn ω)
26		eqid 2229	. . . . 5 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
27	19, 26	fnmpti 5452	. . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
28		eqfnfv 5732	. . . 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 713   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∅c0 3491  ifcif 3602  {csn 3666  {cpr 3667   ↦ cmpt 4145  ωcom 4682   × cxp 4717   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  1_oc1o 6555  2_oc2o 6556  ℕ_∞xnninf 7286

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1o 6562  df-2o 6563  df-map 6797  df-nninf 7287

This theorem is referenced by: (None)