Theorem nninfinfwlpo 7343

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 7296). (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 6815	. . . . . 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 5635	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2766	. . . . . . . 8 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4492	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6	5	rexeqdv 2735	. . . . . . . 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 3624	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑘(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4179	. . . . 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 2230	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → 1o = 1o)
13	12	cbvmptv 4179	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o)
14	13	a1i 9	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑖 ∈ ω ↦ 1o) = (𝑘 ∈ ω ↦ 1o))
15	11, 14	eqeq12d 2244	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ 𝑧 = (𝑘 ∈ ω ↦ 1o)))
16	15	dcbid 843	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ DECID 𝑧 = (𝑘 ∈ ω ↦ 1o)))
17	16	cbvralv 2765	. . . . . 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 7341	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
20	19	ralrimiva 2603	. . 3 ⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
21		omex 4684	. . . 4 ⊢ ω ∈ V
22		iswomnimap 7329	. . . 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 7334	. . 3 ⊢ ((ω ∈ WOmni ∧ 𝑥 ∈ ℕ∞) → DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))
28	27	ralrimiva 2603	. 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 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799  ∅c0 3491  ifcif 3602   ↦ cmpt 4144  suc csuc 4455  ωcom 4681  ⟶wf 5313  ‘cfv 5317  (class class class)co 6000  1_oc1o 6553  2_oc2o 6554   ↑_𝑚 cmap 6793  ℕ_∞xnninf 7282  WOmnicwomni 7326

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1o 6560  df-2o 6561  df-er 6678  df-map 6795  df-en 6886  df-fin 6888  df-nninf 7283  df-womni 7327

This theorem is referenced by: (None)