Theorem nninfwlpoim 7239

Description: Decidable equality for ℕ_∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)

Assertion

Ref	Expression
nninfwlpoim	⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni)

Distinct variable group:   𝑥,𝑦

Proof of Theorem nninfwlpoim

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

Step	Hyp	Ref	Expression
1		elmapi 6726	. . . . 5 ⊢ (𝑓 ∈ (2o ↑𝑚 ω) → 𝑓:ω⟶2o)
2	1	adantl 277	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → 𝑓:ω⟶2o)
3		fveqeq2 5564	. . . . . . . 8 ⊢ (𝑞 = 𝑧 → ((𝑓‘𝑞) = ∅ ↔ (𝑓‘𝑧) = ∅))
4	3	cbvrexv 2727	. . . . . . 7 ⊢ (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅)
5		suceq 4434	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → suc 𝑗 = suc 𝑖)
6	5	rexeqdv 2697	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (∃𝑧 ∈ suc 𝑗(𝑓‘𝑧) = ∅ ↔ ∃𝑧 ∈ suc 𝑖(𝑓‘𝑧) = ∅))
7	4, 6	bitrid 192	. . . . . 6 ⊢ (𝑗 = 𝑖 → (∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅ ↔ ∃𝑧 ∈ suc 𝑖(𝑓‘𝑧) = ∅))
8	7	ifbid 3579	. . . . 5 ⊢ (𝑗 = 𝑖 → if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o) = if(∃𝑧 ∈ suc 𝑖(𝑓‘𝑧) = ∅, ∅, 1o))
9	8	cbvmptv 4126	. . . 4 ⊢ (𝑗 ∈ ω ↦ if(∃𝑞 ∈ suc 𝑗(𝑓‘𝑞) = ∅, ∅, 1o)) = (𝑖 ∈ ω ↦ if(∃𝑧 ∈ suc 𝑖(𝑓‘𝑧) = ∅, ∅, 1o))
10		simpl 109	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)
11		equequ1 1723	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
12	11	dcbid 839	. . . . . 6 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
13		equequ2 1724	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 = 𝑦 ↔ 𝑧 = 𝑤))
14	13	dcbid 839	. . . . . 6 ⊢ (𝑦 = 𝑤 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 𝑤))
15	12, 14	cbvral2v 2739	. . . . 5 ⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℕ∞ ∀𝑤 ∈ ℕ∞ DECID 𝑧 = 𝑤)
16	10, 15	sylib 122	. . . 4 ⊢ ((∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → ∀𝑧 ∈ ℕ∞ ∀𝑤 ∈ ℕ∞ DECID 𝑧 = 𝑤)
17	2, 9, 16	nninfwlpoimlemdc 7238	. . 3 ⊢ ((∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ (2o ↑𝑚 ω)) → DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
18	17	ralrimiva 2567	. 2 ⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
19		omex 4626	. . 3 ⊢ ω ∈ V
20		iswomnimap 7227	. . 3 ⊢ (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o))
21	19, 20	ax-mp 5	. 2 ⊢ (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑛 ∈ ω (𝑓‘𝑛) = 1o)
22	18, 21	sylibr 134	1 ⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760  ∅c0 3447  ifcif 3558   ↦ cmpt 4091  suc csuc 4397  ωcom 4623  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  2_oc2o 6465   ↑_𝑚 cmap 6704  ℕ_∞xnninf 7180  WOmnicwomni 7224

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1o 6471  df-2o 6472  df-er 6589  df-map 6706  df-en 6797  df-fin 6799  df-nninf 7181  df-womni 7225

This theorem is referenced by:  nninfwlpo  7240