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Theorem nndceq0 4531
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 2146 . . . 4 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
21notbid 656 . . . 4 (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 782 . . 3 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4 eqeq1 2146 . . . 4 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
54notbid 656 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
64, 5orbi12d 782 . . 3 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7 eqeq1 2146 . . . 4 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
87notbid 656 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
97, 8orbi12d 782 . . 3 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10 eqeq1 2146 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
1110notbid 656 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
1210, 11orbi12d 782 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13 eqid 2139 . . . 4 ∅ = ∅
1413orci 720 . . 3 (∅ = ∅ ∨ ¬ ∅ = ∅)
15 peano3 4510 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
1615neneqd 2329 . . . . 5 (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
1716olcd 723 . . . 4 (𝑦 ∈ ω → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))
1817a1d 22 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
193, 6, 9, 12, 14, 18finds 4514 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20 df-dc 820 . 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 697  DECID wdc 819   = wceq 1331  wcel 1480  c0 3363  suc csuc 4287  ωcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505
This theorem is referenced by:  omp1eomlem  6979  ctmlemr  6993  elni2  7134  indpi  7162  nnsf  13285  peano4nninf  13286
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