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Theorem nndceq 6191
Description: Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4393. (Contributed by Jim Kingdon, 31-Aug-2019.)
Assertion
Ref Expression
nndceq ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)

Proof of Theorem nndceq
StepHypRef Expression
1 nntri3or 6185 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 elirr 4319 . . . . . . 7 ¬ 𝐴𝐴
3 eleq2 2146 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
42, 3mtbii 632 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
54con2i 590 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65olcd 686 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
7 orc 666 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
8 elirr 4319 . . . . . . 7 ¬ 𝐵𝐵
9 eleq2 2146 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
108, 9mtbiri 633 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
1110con2i 590 . . . . 5 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1211olcd 686 . . . 4 (𝐵𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
136, 7, 123jaoi 1235 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
141, 13syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
15 df-dc 777 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1614, 15sylibr 132 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
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 4367
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 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
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 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368
This theorem is referenced by:  nndifsnid  6195  fidceq  6514  unsnfidcex  6556  unsnfidcel  6557  enqdc  6822
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