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Theorem nndcel 6144
Description: Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
Assertion
Ref Expression
nndcel ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)

Proof of Theorem nndcel
StepHypRef Expression
1 nntri3or 6137 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 orc 666 . . . 4 (𝐴𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
3 elirr 4292 . . . . . 6 ¬ 𝐵𝐵
4 eleq1 2142 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
53, 4mtbiri 633 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
65olcd 686 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
7 en2lp 4305 . . . . . 6 ¬ (𝐵𝐴𝐴𝐵)
87imnani 658 . . . . 5 (𝐵𝐴 → ¬ 𝐴𝐵)
98olcd 686 . . . 4 (𝐵𝐴 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
102, 6, 93jaoi 1235 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
111, 10syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
12 df-dc 777 . 2 (DECID 𝐴𝐵 ↔ (𝐴𝐵 ∨ ¬ 𝐴𝐵))
1311, 12sylibr 132 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 776  w3o 919   = wceq 1285  wcel 1434  ωcom 4339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340
This theorem is referenced by:  ltdcpi  6575
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