Theorem nninfinfwlpo 7484

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 7437). (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 6917	. . . . . 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 5684	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2781	. . . . . . . 8 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4528	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6	5	rexeqdv 2750	. . . . . . . 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 3648	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4211	. . . . 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 2235	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → 1o = 1o)
13	12	cbvmptv 4211	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o)
14	13	a1i 9	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o))
15	11, 14	eqeq12d 2249	. . . . . . . 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 2780	. . . . . 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 7482	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
20	19	ralrimiva 2617	. . 3 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
21		omex 4720	. . . 4 ⊢ ω ∈ V
22		iswomnimap 7470	. . . 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 7475	. . 3 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
28	27	ralrimiva 2617	. 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 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815  ∅c0 3512  ifcif 3624   ↦ cmpt 4176  suc csuc 4491  ωcom 4717  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  2_oc2o 6654   ↑_𝑚 cmap 6895  ℕ_∞xnninf 7423  WOmnicwomni 7467

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-coll 4230  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-reu 2529  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-iun 3998  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-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1o 6660  df-2o 6661  df-er 6780  df-map 6897  df-en 6989  df-fin 6991  df-nninf 7424  df-womni 7468

This theorem is referenced by: (None)