Theorem nninfwlpor 7235

Description: The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ_∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem nninfwlpor

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninff 7183	. . . 4 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
2	1	ad2antrl 490	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑥:ω⟶2o)
3		nninff 7183	. . . 4 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
4	3	ad2antll 491	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑦:ω⟶2o)
5		fveq2 5555	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑥‘𝑗) = (𝑥‘𝑖))
6		fveq2 5555	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑦‘𝑗) = (𝑦‘𝑖))
7	5, 6	eqeq12d 2208	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑥‘𝑗) = (𝑦‘𝑗) ↔ (𝑥‘𝑖) = (𝑦‘𝑖)))
8	7	ifbid 3579	. . . 4 ⊢ (𝑗 = 𝑖 → if((𝑥‘𝑗) = (𝑦‘𝑗), 1o, ∅) = if((𝑥‘𝑖) = (𝑦‘𝑖), 1o, ∅))
9	8	cbvmptv 4126	. . 3 ⊢ (𝑗 ∈ ω ↦ if((𝑥‘𝑗) = (𝑦‘𝑗), 1o, ∅)) = (𝑖 ∈ ω ↦ if((𝑥‘𝑖) = (𝑦‘𝑖), 1o, ∅))
10		simpl 109	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → ω ∈ WOmni)
11	2, 4, 9, 10	nninfwlporlem 7234	. 2 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → DECID 𝑥 = 𝑦)
12	11	ralrimivva 2576	1 ⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∅c0 3447  ifcif 3558   ↦ cmpt 4091  ωcom 4623  ⟶wf 5251  ‘cfv 5255  1_oc1o 6464  2_oc2o 6465  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-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-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-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-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1o 6471  df-2o 6472  df-map 6706  df-nninf 7181  df-womni 7225

This theorem is referenced by:  nninfwlpo  7240