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Theorem nndceq 6395
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 4531. (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 6389 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 elirr 4456 . . . . . . 7 ¬ 𝐴𝐴
3 eleq2 2203 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
42, 3mtbii 663 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
54con2i 616 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65olcd 723 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
7 orc 701 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
8 elirr 4456 . . . . . . 7 ¬ 𝐵𝐵
9 eleq2 2203 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
108, 9mtbiri 664 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
1110con2i 616 . . . . 5 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1211olcd 723 . . . 4 (𝐵𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
136, 7, 123jaoi 1281 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
141, 13syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
15 df-dc 820 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1614, 15sylibr 133 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819  w3o 961   = wceq 1331  wcel 1480  ωcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505
This theorem is referenced by:  nndifsnid  6403  fidceq  6763  unsnfidcex  6808  unsnfidcel  6809  enqdc  7181  nninfsellemdc  13267
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