Theorem nninfwlpor 7341

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 7289	. . . 4 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
2	1	ad2antrl 490	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑥:ω⟶2o)
3		nninff 7289	. . . 4 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
4	3	ad2antll 491	. . 3 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → 𝑦:ω⟶2o)
5		fveq2 5627	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑥‘𝑗) = (𝑥‘𝑖))
6		fveq2 5627	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑦‘𝑗) = (𝑦‘𝑖))
7	5, 6	eqeq12d 2244	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑥‘𝑗) = (𝑦‘𝑗) ↔ (𝑥‘𝑖) = (𝑦‘𝑖)))
8	7	ifbid 3624	. . . 4 ⊢ (𝑗 = 𝑖 → if((𝑥‘𝑗) = (𝑦‘𝑗), 1o, ∅) = if((𝑥‘𝑖) = (𝑦‘𝑖), 1o, ∅))
9	8	cbvmptv 4180	. . 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 7340	. 2 ⊢ ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞)) → DECID 𝑥 = 𝑦)
12	11	ralrimivva 2612	1 ⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∅c0 3491  ifcif 3602   ↦ cmpt 4145  ωcom 4682  ⟶wf 5314  ‘cfv 5318  1_oc1o 6555  2_oc2o 6556  ℕ_∞xnninf 7286  WOmnicwomni 7330

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-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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-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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1o 6562  df-2o 6563  df-map 6797  df-nninf 7287  df-womni 7331

This theorem is referenced by:  nninfwlpo  7348