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Theorem nndceq 6300
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 4459. (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 6294 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 elirr 4385 . . . . . . 7 ¬ 𝐴𝐴
3 eleq2 2158 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
42, 3mtbii 637 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
54con2i 595 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65olcd 691 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
7 orc 671 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
8 elirr 4385 . . . . . . 7 ¬ 𝐵𝐵
9 eleq2 2158 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
108, 9mtbiri 638 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
1110con2i 595 . . . . 5 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1211olcd 691 . . . 4 (𝐵𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
136, 7, 123jaoi 1246 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
141, 13syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
15 df-dc 784 . 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 667  DECID wdc 783  w3o 926   = wceq 1296  wcel 1445  ωcom 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434
This theorem is referenced by:  nndifsnid  6306  fidceq  6665  unsnfidcex  6710  unsnfidcel  6711  enqdc  7017  nninfsellemdc  12616
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