Theorem nninfnfiinf 16625

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 16624	. . . . . 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 7320	. . . . . . . . . 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 5783	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ 2o)
11		df2o3 6596	. . . . . . . 8 ⊢ 2o = {∅, 1o}
12	10, 11	eleqtrdi 2324	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) ∈ {∅, 1o})
13		elpri 3692	. . . . . . 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 736	. . . . 5 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → ((𝐴‘𝑗) = 1o ∨ (𝐴‘𝑗) = ∅))
16	6, 15	ecased 1385	. . . 4 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = 1o)
17		fconstmpt 4773	. . . . . . 7 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
18	17	fveq1i 5640	. . . . . 6 ⊢ ((ω × {1o})‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)
19		1oex 6589	. . . . . . 7 ⊢ 1o ∈ V
20	19	fvconst2 5869	. . . . . 6 ⊢ (𝑗 ∈ ω → ((ω × {1o})‘𝑗) = 1o)
21	18, 20	eqtr3id 2278	. . . . 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 2267	. . 3 ⊢ (((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ∧ 𝑗 ∈ ω) → (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	23	ralrimiva 2605	. 2 ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑗 ∈ ω (𝐴‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
25	7	ffnd 5483	. . . 4 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 Fn ω)
26		eqid 2231	. . . . 5 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
27	19, 26	fnmpti 5461	. . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
28		eqfnfv 5744	. . . 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 715   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  ∅c0 3494  ifcif 3605  {csn 3669  {cpr 3670   ↦ cmpt 4150  ωcom 4688   × cxp 4723   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575  ℕ_∞xnninf 7317

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-map 6818  df-nninf 7318

This theorem is referenced by: (None)