ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nndceq0 GIF version

Theorem nndceq0 4469
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
nndceq0 (𝐴 ∈ ω → DECID 𝐴 = ∅)

Proof of Theorem nndceq0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2106 . . . 4 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
21notbid 633 . . . 4 (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 748 . . 3 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4 eqeq1 2106 . . . 4 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
54notbid 633 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
64, 5orbi12d 748 . . 3 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7 eqeq1 2106 . . . 4 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
87notbid 633 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
97, 8orbi12d 748 . . 3 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10 eqeq1 2106 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
1110notbid 633 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
1210, 11orbi12d 748 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13 eqid 2100 . . . 4 ∅ = ∅
1413orci 691 . . 3 (∅ = ∅ ∨ ¬ ∅ = ∅)
15 peano3 4448 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
1615neneqd 2288 . . . . 5 (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
1716olcd 694 . . . 4 (𝑦 ∈ ω → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))
1817a1d 22 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
193, 6, 9, 12, 14, 18finds 4452 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20 df-dc 787 . 2 (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
2119, 20sylibr 133 1 (𝐴 ∈ ω → DECID 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 670  DECID wdc 786   = wceq 1299  wcel 1448  c0 3310  suc csuc 4225  ωcom 4442
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443
This theorem is referenced by:  omp1eomlem  6894  ctmlemr  6908  elni2  7023  indpi  7051  nnsf  12783  peano4nninf  12784
  Copyright terms: Public domain W3C validator