Theorem nninfinfwlpo 7378

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 7331). (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 6838	. . . . . 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 5648	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2768	. . . . . . . 8 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4499	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6	5	rexeqdv 2737	. . . . . . . 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 3627	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4185	. . . . 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 2232	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → 1o = 1o)
13	12	cbvmptv 4185	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o)
14	13	a1i 9	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o))
15	11, 14	eqeq12d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ 𝑧 = (𝑘 ∈ ω ↦ 1o)))
16	15	dcbid 845	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o)))
17	16	cbvralv 2767	. . . . . 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 7376	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
20	19	ralrimiva 2605	. . 3 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
21		omex 4691	. . . 4 ⊢ ω ∈ V
22		iswomnimap 7364	. . . 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 7369	. . 3 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
28	27	ralrimiva 2605	. 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  suc csuc 4462  ωcom 4688  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  1_oc1o 6574  2_oc2o 6575   ↑_𝑚 cmap 6816  ℕ_∞xnninf 7317  WOmnicwomni 7361

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-coll 4204  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-reu 2517  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-iun 3972  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-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-er 6701  df-map 6818  df-en 6909  df-fin 6911  df-nninf 7318  df-womni 7362

This theorem is referenced by: (None)