Theorem nndceq0 4667

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 2212	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2	1	notbid 669	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅))
3	1, 2	orbi12d 795	. . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅)))
4		eqeq1 2212	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5	4	notbid 669	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅))
6	4, 5	orbi12d 795	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)))
7		eqeq1 2212	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8	7	notbid 669	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅))
9	7, 8	orbi12d 795	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅)))
10		eqeq1 2212	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11	10	notbid 669	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅))
12	10, 11	orbi12d 795	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)))
13		eqid 2205	. . . 4 ⊢ ∅ = ∅
14	13	orci 733	. . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅)
15		peano3 4645	. . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅)
16	15	neneqd 2397	. . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅)
17	16	olcd 736	. . . 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 4649	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
20		df-dc 837	. 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))
21	19, 20	sylibr 134	1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2176  ∅c0 3460  suc csuc 4413  ωcom 4639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4419  df-iom 4640

This theorem is referenced by:  omp1eomlem  7198  ctmlemr  7212  nnnninfeq2  7233  nninfisol  7237  elni2  7429  indpi  7457  nnsf  15979  peano4nninf  15980