Theorem nninfwlpor 7276

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 7224	. . . 4 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
2	1	ad2antrl 490	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑥:ω⟶2o)
3		nninff 7224	. . . 4 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
4	3	ad2antll 491	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑦:ω⟶2o)
5		fveq2 5576	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑥‘𝑗) = (𝑥‘𝑖))
6		fveq2 5576	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑦‘𝑗) = (𝑦‘𝑖))
7	5, 6	eqeq12d 2220	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑥‘𝑗) = (𝑦‘𝑗) ↔ (𝑥‘𝑖) = (𝑦‘𝑖)))
8	7	ifbid 3592	. . . 4 ⊢ (𝑗 = 𝑖 → if((𝑥‘𝑗) = (𝑦‘𝑗), 1o, ∅) = if((𝑥‘𝑖) = (𝑦‘𝑖), 1o, ∅))
9	8	cbvmptv 4140	. . 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 7275	. 2 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → DECID 𝑥 = 𝑦)
12	11	ralrimivva 2588	1 ⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 836   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ∅c0 3460  ifcif 3571   ↦ cmpt 4105  ωcom 4638  ⟶wf 5267  ‘cfv 5271  1_oc1o 6495  2_oc2o 6496  ℕ_∞xnninf 7221  WOmnicwomni 7265

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1o 6502  df-2o 6503  df-map 6737  df-nninf 7222  df-womni 7266

This theorem is referenced by:  nninfwlpo  7283