Theorem nndceq0 4602

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 2177	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2	1	notbid 662	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
3	1, 2	orbi12d 788	. . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4		eqeq1 2177	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5	4	notbid 662	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
6	4, 5	orbi12d 788	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7		eqeq1 2177	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8	7	notbid 662	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
9	7, 8	orbi12d 788	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10		eqeq1 2177	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11	10	notbid 662	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
12	10, 11	orbi12d 788	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13		eqid 2170	. . . 4 ⊢ ∅ = ∅
14	13	orci 726	. . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅)
15		peano3 4580	. . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
16	15	neneqd 2361	. . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
17	16	olcd 729	. . . 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 4584	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20		df-dc 830	. 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
21	19, 20	sylibr 133	1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∅c0 3414  suc csuc 4350  ωcom 4574

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 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

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 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575

This theorem is referenced by:  omp1eomlem  7071  ctmlemr  7085  nnnninfeq2  7105  nninfisol  7109  elni2  7276  indpi  7304  nnsf  14038  peano4nninf  14039