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Theorem nndceq 6525
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 4635. (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 6519 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 elirr 4558 . . . . . . 7 ¬ 𝐴𝐴
3 eleq2 2253 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
42, 3mtbii 675 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
54con2i 628 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65olcd 735 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
7 orc 713 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
8 elirr 4558 . . . . . . 7 ¬ 𝐵𝐵
9 eleq2 2253 . . . . . . 7 (𝐴 = 𝐵 → (𝐵𝐴𝐵𝐵))
108, 9mtbiri 676 . . . . . 6 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
1110con2i 628 . . . . 5 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1211olcd 735 . . . 4 (𝐵𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
136, 7, 123jaoi 1314 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
141, 13syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
15 df-dc 836 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1614, 15sylibr 134 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  DECID wdc 835  w3o 979   = wceq 1364  wcel 2160  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608
This theorem is referenced by:  nndifsnid  6533  fidceq  6898  unsnfidcex  6949  unsnfidcel  6950  nninfwlporlemd  7201  nninfwlporlem  7202  nninfwlpoimlemg  7204  nninfwlpoimlemginf  7205  2onetap  7285  2omotaplemap  7287  enqdc  7391  xpscf  12826  nninfsellemdc  15238
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