Theorem nndceq0 4357

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.

Step	Hyp	Ref	Expression
1		eqeq1 2087	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2	1	notbid 624	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
3	1, 2	orbi12d 739	. . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4		eqeq1 2087	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5	4	notbid 624	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
6	4, 5	orbi12d 739	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7		eqeq1 2087	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8	7	notbid 624	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
9	7, 8	orbi12d 739	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10		eqeq1 2087	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11	10	notbid 624	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
12	10, 11	orbi12d 739	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13		eqid 2081	. . . 4 ⊢ ∅ = ∅
14	13	orci 682	. . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅)
15		peano3 4337	. . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
16	15	neneqd 2266	. . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
17	16	olcd 685	. . . 4 ⊢ (𝑦 ∈ ω → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))
18	17	a1d 22	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
19	3, 6, 9, 12, 14, 18	finds 4341	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20		df-dc 776	. 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
21	19, 20	sylibr 132	1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775   = wceq 1284   ∈ wcel 1433  ∅c0 3251  suc csuc 4120  ωcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332

This theorem is referenced by:  elni2  6504  indpi  6532