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Theorem nndc 4235
 Description: A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
Assertion
Ref Expression
nndc (A 𝜔 → DECID A = ∅)

Proof of Theorem nndc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2028 . . . 4 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
21notbid 579 . . . 4 (x = ∅ → (¬ x = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 695 . . 3 (x = ∅ → ((x = ∅ ¬ x = ∅) ↔ (∅ = ∅ ¬ ∅ = ∅)))
4 eqeq1 2028 . . . 4 (x = y → (x = ∅ ↔ y = ∅))
54notbid 579 . . . 4 (x = y → (¬ x = ∅ ↔ ¬ y = ∅))
64, 5orbi12d 695 . . 3 (x = y → ((x = ∅ ¬ x = ∅) ↔ (y = ∅ ¬ y = ∅)))
7 eqeq1 2028 . . . 4 (x = suc y → (x = ∅ ↔ suc y = ∅))
87notbid 579 . . . 4 (x = suc y → (¬ x = ∅ ↔ ¬ suc y = ∅))
97, 8orbi12d 695 . . 3 (x = suc y → ((x = ∅ ¬ x = ∅) ↔ (suc y = ∅ ¬ suc y = ∅)))
10 eqeq1 2028 . . . 4 (x = A → (x = ∅ ↔ A = ∅))
1110notbid 579 . . . 4 (x = A → (¬ x = ∅ ↔ ¬ A = ∅))
1210, 11orbi12d 695 . . 3 (x = A → ((x = ∅ ¬ x = ∅) ↔ (A = ∅ ¬ A = ∅)))
13 eqid 2022 . . . 4 ∅ = ∅
1413orci 637 . . 3 (∅ = ∅ ¬ ∅ = ∅)
15 peano3 4215 . . . . . 6 (y 𝜔 → suc y ≠ ∅)
1615neneqd 2203 . . . . 5 (y 𝜔 → ¬ suc y = ∅)
1716olcd 640 . . . 4 (y 𝜔 → (suc y = ∅ ¬ suc y = ∅))
1817a1d 22 . . 3 (y 𝜔 → ((y = ∅ ¬ y = ∅) → (suc y = ∅ ¬ suc y = ∅)))
193, 6, 9, 12, 14, 18finds 4219 . 2 (A 𝜔 → (A = ∅ ¬ A = ∅))
20 df-dc 735 . 2 (DECID A = ∅ ↔ (A = ∅ ¬ A = ∅))
2119, 20sylibr 137 1 (A 𝜔 → DECID A = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  DECID wdc 734   = wceq 1373   ∈ wcel 1375  ∅c0 3202  suc csuc 4026  𝜔com 4209 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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-nul 3835  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-iinf 4207 This theorem depends on definitions:  df-bi 110  df-dc 735  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-int 3568  df-suc 4031  df-iom 4210 This theorem is referenced by: (None)
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