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Theorem nndceq0 4266
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 2028 . . . 4 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
21notbid 579 . . . 4 (x = ∅ → (¬ x = ∅ ↔ ¬ ∅ = ∅))
31, 2orbi12d 694 . . 3 (x = ∅ → ((x = ∅ ¬ x = ∅) ↔ (∅ = ∅ ¬ ∅ = ∅)))
4 eqeq1 2028 . . . 4 (x = y → (x = ∅ ↔ y = ∅))
54notbid 579 . . . 4 (x = y → (¬ x = ∅ ↔ ¬ y = ∅))
64, 5orbi12d 694 . . 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 694 . . 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 694 . . 3 (x = A → ((x = ∅ ¬ x = ∅) ↔ (A = ∅ ¬ A = ∅)))
13 eqid 2022 . . . 4 ∅ = ∅
1413orci 637 . . 3 (∅ = ∅ ¬ ∅ = ∅)
15 peano3 4246 . . . . . 6 (y 𝜔 → suc y ≠ ∅)
1615neneqd 2205 . . . . 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 4250 . 2 (A 𝜔 → (A = ∅ ¬ A = ∅))
20 df-dc 734 . 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 733   = wceq 1228   wcel 1374  c0 3201  suc csuc 4051  𝜔com 4240
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by:  elni2  6174
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