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Theorem nndceq0 4282
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 2043 . . . 4 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
21notbid 591 . . . 4 (x = ∅ → (¬ x = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 706 . . 3 (x = ∅ → ((x = ∅ ¬ x = ∅) ↔ (∅ = ∅ ¬ ∅ = ∅)))
4 eqeq1 2043 . . . 4 (x = y → (x = ∅ ↔ y = ∅))
54notbid 591 . . . 4 (x = y → (¬ x = ∅ ↔ ¬ y = ∅))
64, 5orbi12d 706 . . 3 (x = y → ((x = ∅ ¬ x = ∅) ↔ (y = ∅ ¬ y = ∅)))
7 eqeq1 2043 . . . 4 (x = suc y → (x = ∅ ↔ suc y = ∅))
87notbid 591 . . . 4 (x = suc y → (¬ x = ∅ ↔ ¬ suc y = ∅))
97, 8orbi12d 706 . . 3 (x = suc y → ((x = ∅ ¬ x = ∅) ↔ (suc y = ∅ ¬ suc y = ∅)))
10 eqeq1 2043 . . . 4 (x = A → (x = ∅ ↔ A = ∅))
1110notbid 591 . . . 4 (x = A → (¬ x = ∅ ↔ ¬ A = ∅))
1210, 11orbi12d 706 . . 3 (x = A → ((x = ∅ ¬ x = ∅) ↔ (A = ∅ ¬ A = ∅)))
13 eqid 2037 . . . 4 ∅ = ∅
1413orci 649 . . 3 (∅ = ∅ ¬ ∅ = ∅)
15 peano3 4262 . . . . . 6 (y 𝜔 → suc y ≠ ∅)
1615neneqd 2221 . . . . 5 (y 𝜔 → ¬ suc y = ∅)
1716olcd 652 . . . 4 (y 𝜔 → (suc y = ∅ ¬ suc y = ∅))
1817a1d 22 . . 3 (y 𝜔 → ((y = ∅ ¬ y = ∅) → (suc y = ∅ ¬ suc y = ∅)))
193, 6, 9, 12, 14, 18finds 4266 . 2 (A 𝜔 → (A = ∅ ¬ A = ∅))
20 df-dc 742 . 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 628  DECID wdc 741   = wceq 1242   wcel 1390  c0 3218  suc csuc 4068  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257
This theorem is referenced by:  elni2  6298  indpi  6326
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