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Theorem nndceq 5990
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 4266. (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 5987 . . 3 ((A 𝜔 B 𝜔) → (A B A = B B A))
2 elirr 4208 . . . . . . 7 ¬ A A
3 eleq2 2083 . . . . . . 7 (A = B → (A AA B))
42, 3mtbii 586 . . . . . 6 (A = B → ¬ A B)
54con2i 545 . . . . 5 (A B → ¬ A = B)
65olcd 640 . . . 4 (A B → (A = B ¬ A = B))
7 orc 620 . . . 4 (A = B → (A = B ¬ A = B))
8 elirr 4208 . . . . . . 7 ¬ B B
9 eleq2 2083 . . . . . . 7 (A = B → (B AB B))
108, 9mtbiri 587 . . . . . 6 (A = B → ¬ B A)
1110con2i 545 . . . . 5 (B A → ¬ A = B)
1211olcd 640 . . . 4 (B A → (A = B ¬ A = B))
136, 7, 123jaoi 1184 . . 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 734 . 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 616  DECID wdc 733   w3o 872   = wceq 1228   wcel 1374  𝜔com 4240
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241
This theorem is referenced by:  enqdc  6220
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