Theorem nninfinfwlpo 7471

Description: The point at infinity in ℕ_∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ_∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ_∞ corresponding to natural numbers are isolated (nninfisol 7424). (Contributed by Jim Kingdon, 25-Nov-2025.)

Assertion

Ref	Expression
nninfinfwlpo	⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)

Distinct variable group:   𝑥,𝑖

Proof of Theorem nninfinfwlpo

Dummy variables 𝑓 𝑘 𝑛 𝑧 𝑗 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 6904	. . . . . 6 ⊢ (𝑓 ∈ (2o ↑𝑚 ω) → 𝑓:ω⟶2o)
2	1	adantl 277	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → 𝑓:ω⟶2o)
3		fveqeq2 5679	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2779	. . . . . . . 8 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4523	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6	5	rexeqdv 2748	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅ ↔ ∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅))
7	4, 6	bitrid 192	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅))
8	7	ifbid 3644	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4206	. . . . 5 ⊢ (𝑗 ∈ ω ↦ if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o)) = (𝑘 ∈ ω ↦ if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
10		simpl 109	. . . . . 6 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → ∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
11		id 19	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
12		eqidd 2233	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → 1o = 1o)
13	12	cbvmptv 4206	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o)
14	13	a1i 9	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o))
15	11, 14	eqeq12d 2247	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ 𝑧 = (𝑘 ∈ ω ↦ 1o)))
16	15	dcbid 846	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o)))
17	16	cbvralv 2778	. . . . . 6 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑧 ∈ ℕ∞ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o))
18	10, 17	sylib 122	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → ∀𝑧 ∈ ℕ∞ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o))
19	2, 9, 18	nninfinfwlpolem 7469	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
20	19	ralrimiva 2615	. . 3 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
21		omex 4715	. . . 4 ⊢ ω ∈ V
22		iswomnimap 7457	. . . 4 ⊢ (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o))
23	21, 22	ax-mp 5	. . 3 ⊢ (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
24	20, 23	sylibr 134	. 2 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ω ∈ WOmni)
25		simpl 109	. . . 4 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → ω ∈ WOmni)
26		simpr 110	. . . 4 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → 𝑥 ∈ ℕ∞)
27	25, 26	nninfdcinf 7462	. . 3 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
28	27	ralrimiva 2615	. 2 ⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
29	24, 28	impbii 126	1 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)

Syntax hints:   ∧ wa 104   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2813  ∅c0 3508  ifcif 3620   ↦ cmpt 4171  suc csuc 4486  ωcom 4712  ⟶wf 5348  ‘cfv 5352  (class class class)co 6050  1_oc1o 6640  2_oc2o 6641   ↑_𝑚 cmap 6882  ℕ_∞xnninf 7410  WOmnicwomni 7454

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1o 6647  df-2o 6648  df-er 6767  df-map 6884  df-en 6976  df-fin 6978  df-nninf 7411  df-womni 7455

This theorem is referenced by: (None)