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Theorem nndcel 6078
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 6072 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 orc 633 . . . 4 (𝐴𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
3 elirr 4266 . . . . . 6 ¬ 𝐵𝐵
4 eleq1 2100 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
53, 4mtbiri 600 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
65olcd 653 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
7 en2lp 4278 . . . . . 6 ¬ (𝐵𝐴𝐴𝐵)
87imnani 625 . . . . 5 (𝐵𝐴 → ¬ 𝐴𝐵)
98olcd 653 . . . 4 (𝐵𝐴 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
102, 6, 93jaoi 1198 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
111, 10syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
12 df-dc 743 . 2 (DECID 𝐴𝐵 ↔ (𝐴𝐵 ∨ ¬ 𝐴𝐵))
1311, 12sylibr 137 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wo 629  DECID wdc 742  w3o 884   = wceq 1243  wcel 1393  ωcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314
This theorem is referenced by:  ltdcpi  6421
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