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Theorem nninfwlpor 7175
Description: The Weak Limited Principle of Omniscience (WLPO) implies that equality for is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
Assertion
Ref Expression
nninfwlpor (ω ∈ WOmni → ∀𝑥 ∈ ℕ𝑦 ∈ 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.
StepHypRef Expression
1 nninff 7124 . . . 4 (𝑥 ∈ ℕ𝑥:ω⟶2o)
21ad2antrl 490 . . 3 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → 𝑥:ω⟶2o)
3 nninff 7124 . . . 4 (𝑦 ∈ ℕ𝑦:ω⟶2o)
43ad2antll 491 . . 3 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → 𝑦:ω⟶2o)
5 fveq2 5517 . . . . . 6 (𝑗 = 𝑖 → (𝑥𝑗) = (𝑥𝑖))
6 fveq2 5517 . . . . . 6 (𝑗 = 𝑖 → (𝑦𝑗) = (𝑦𝑖))
75, 6eqeq12d 2192 . . . . 5 (𝑗 = 𝑖 → ((𝑥𝑗) = (𝑦𝑗) ↔ (𝑥𝑖) = (𝑦𝑖)))
87ifbid 3557 . . . 4 (𝑗 = 𝑖 → if((𝑥𝑗) = (𝑦𝑗), 1o, ∅) = if((𝑥𝑖) = (𝑦𝑖), 1o, ∅))
98cbvmptv 4101 . . 3 (𝑗 ∈ ω ↦ if((𝑥𝑗) = (𝑦𝑗), 1o, ∅)) = (𝑖 ∈ ω ↦ if((𝑥𝑖) = (𝑦𝑖), 1o, ∅))
10 simpl 109 . . 3 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → ω ∈ WOmni)
112, 4, 9, 10nninfwlporlem 7174 . 2 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → DECID 𝑥 = 𝑦)
1211ralrimivva 2559 1 (ω ∈ WOmni → ∀𝑥 ∈ ℕ𝑦 ∈ ℕ DECID 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  c0 3424  ifcif 3536  cmpt 4066  ωcom 4591  wf 5214  cfv 5218  1oc1o 6413  2oc2o 6414  xnninf 7121  WOmnicwomni 7164
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1o 6420  df-2o 6421  df-map 6653  df-nninf 7122  df-womni 7165
This theorem is referenced by:  nninfwlpo  7180
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