Theorem nninfwlpor 7433

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 7381	. . . 4 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
2	1	ad2antrl 490	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑥:ω⟶2o)
3		nninff 7381	. . . 4 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
4	3	ad2antll 491	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑦:ω⟶2o)
5		fveq2 5648	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑥‘𝑗) = (𝑥‘𝑖))
6		fveq2 5648	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑦‘𝑗) = (𝑦‘𝑖))
7	5, 6	eqeq12d 2246	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑥‘𝑗) = (𝑦‘𝑗) ↔ (𝑥‘𝑖) = (𝑦‘𝑖)))
8	7	ifbid 3631	. . . 4 ⊢ (𝑗 = 𝑖 → if((𝑥‘𝑗) = (𝑦‘𝑗), 1o, ∅) = if((𝑥‘𝑖) = (𝑦‘𝑖), 1o, ∅))
9	8	cbvmptv 4190	. . 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 7432	. 2 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → DECID 𝑥 = 𝑦)
12	11	ralrimivva 2615	1 ⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∅c0 3496  ifcif 3607   ↦ cmpt 4155  ωcom 4694  ⟶wf 5329  ‘cfv 5333  1_oc1o 6618  2_oc2o 6619  ℕ_∞xnninf 7378  WOmnicwomni 7422

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-map 6862  df-nninf 7379  df-womni 7423

This theorem is referenced by:  nninfwlpo  7440