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Theorem nninfwlpor 7169
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 7118 . . . 4 (𝑥 ∈ ℕ𝑥:ω⟶2o)
21ad2antrl 490 . . 3 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → 𝑥:ω⟶2o)
3 nninff 7118 . . . 4 (𝑦 ∈ ℕ𝑦:ω⟶2o)
43ad2antll 491 . . 3 ((ω ∈ WOmni ∧ (𝑥 ∈ ℕ𝑦 ∈ ℕ)) → 𝑦:ω⟶2o)
5 fveq2 5514 . . . . . 6 (𝑗 = 𝑖 → (𝑥𝑗) = (𝑥𝑖))
6 fveq2 5514 . . . . . 6 (𝑗 = 𝑖 → (𝑦𝑗) = (𝑦𝑖))
75, 6eqeq12d 2192 . . . . 5 (𝑗 = 𝑖 → ((𝑥𝑗) = (𝑦𝑗) ↔ (𝑥𝑖) = (𝑦𝑖)))
87ifbid 3555 . . . 4 (𝑗 = 𝑖 → if((𝑥𝑗) = (𝑦𝑗), 1o, ∅) = if((𝑥𝑖) = (𝑦𝑖), 1o, ∅))
98cbvmptv 4098 . . 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 7168 . 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 3422  ifcif 3534  cmpt 4063  ωcom 4588  wf 5211  cfv 5215  1oc1o 6407  2oc2o 6408  xnninf 7115  WOmnicwomni 7158
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 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1o 6414  df-2o 6415  df-map 6647  df-nninf 7116  df-womni 7159
This theorem is referenced by:  nninfwlpo  7174
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