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Theorem nndceq 6077
 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 4339. (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 6072 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 elirr 4266 . . . . . . 7 ¬ 𝐴𝐴
3 eleq2 2101 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
42, 3mtbii 599 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
54con2i 557 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65olcd 653 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
7 orc 633 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
8 elirr 4266 . . . . . . 7 ¬ 𝐵𝐵
9 eleq2 2101 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
108, 9mtbiri 600 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
1110con2i 557 . . . . 5 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1211olcd 653 . . . 4 (𝐵𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
136, 7, 123jaoi 1198 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
141, 13syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
15 df-dc 743 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1614, 15sylibr 137 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
 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:  nndifsnid  6080  fidceq  6330  enqdc  6459
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