Theorem nninfinfwlpo 7520

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 7473). (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 6944	. . . . . 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 5704	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2787	. . . . . . . 8 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4547	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6	5	rexeqdv 2756	. . . . . . . 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 3662	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4227	. . . . 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 2239	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → 1o = 1o)
13	12	cbvmptv 4227	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o)
14	13	a1i 9	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o))
15	11, 14	eqeq12d 2253	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ 𝑧 = (𝑘 ∈ ω ↦ 1o)))
16	15	dcbid 850	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o)))
17	16	cbvralv 2786	. . . . . 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 7518	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
20	19	ralrimiva 2623	. . 3 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
21		omex 4740	. . . 4 ⊢ ω ∈ V
22		iswomnimap 7506	. . . 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 7511	. . 3 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
28	27	ralrimiva 2623	. 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)

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821  ∅c0 3520  ifcif 3638   ↦ cmpt 4192  suc csuc 4510  ωcom 4737  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1_oc1o 6680  2_oc2o 6681   ↑_𝑚 cmap 6922  ℕ_∞xnninf 7459  WOmnicwomni 7503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-fin 7025  df-nninf 7460  df-womni 7504

This theorem is used by: (None)