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Theorem nndceq0 4600
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 2177 . . . 4 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
21notbid 662 . . . 4 (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 788 . . 3 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4 eqeq1 2177 . . . 4 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
54notbid 662 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
64, 5orbi12d 788 . . 3 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7 eqeq1 2177 . . . 4 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
87notbid 662 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
97, 8orbi12d 788 . . 3 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10 eqeq1 2177 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
1110notbid 662 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
1210, 11orbi12d 788 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13 eqid 2170 . . . 4 ∅ = ∅
1413orci 726 . . 3 (∅ = ∅ ∨ ¬ ∅ = ∅)
15 peano3 4578 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
1615neneqd 2361 . . . . 5 (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
1716olcd 729 . . . 4 (𝑦 ∈ ω → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))
1817a1d 22 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
193, 6, 9, 12, 14, 18finds 4582 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20 df-dc 830 . 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 703  DECID wdc 829   = wceq 1348  wcel 2141  c0 3414  suc csuc 4348  ωcom 4572
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573
This theorem is referenced by:  omp1eomlem  7067  ctmlemr  7081  nnnninfeq2  7101  nninfisol  7105  elni2  7263  indpi  7291  nnsf  13960  peano4nninf  13961
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