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Theorem nndceq 6015
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 4282. (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 6011 . . 3 ((A 𝜔 B 𝜔) → (A B A = B B A))
2 elirr 4224 . . . . . . 7 ¬ A A
3 eleq2 2098 . . . . . . 7 (A = B → (A AA B))
42, 3mtbii 598 . . . . . 6 (A = B → ¬ A B)
54con2i 557 . . . . 5 (A B → ¬ A = B)
65olcd 652 . . . 4 (A B → (A = B ¬ A = B))
7 orc 632 . . . 4 (A = B → (A = B ¬ A = B))
8 elirr 4224 . . . . . . 7 ¬ B B
9 eleq2 2098 . . . . . . 7 (A = B → (B AB B))
108, 9mtbiri 599 . . . . . 6 (A = B → ¬ B A)
1110con2i 557 . . . . 5 (B A → ¬ A = B)
1211olcd 652 . . . 4 (B A → (A = B ¬ A = B))
136, 7, 123jaoi 1197 . . 3 ((A B A = B B A) → (A = B ¬ A = B))
141, 13syl 14 . 2 ((A 𝜔 B 𝜔) → (A = B ¬ A = B))
15 df-dc 742 . 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 628  DECID wdc 741   w3o 883   = wceq 1242   wcel 1390  𝜔com 4256
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257
This theorem is referenced by:  enqdc  6345
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