![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nndceq0 | GIF version |
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.) |
Ref | Expression |
---|---|
nndceq0 | ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2196 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | 1 | notbid 668 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅)) |
3 | 1, 2 | orbi12d 794 | . . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅))) |
4 | eqeq1 2196 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | 4 | notbid 668 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅)) |
6 | 4, 5 | orbi12d 794 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))) |
7 | eqeq1 2196 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | 7 | notbid 668 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅)) |
9 | 7, 8 | orbi12d 794 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))) |
10 | eqeq1 2196 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | 10 | notbid 668 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅)) |
12 | 10, 11 | orbi12d 794 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))) |
13 | eqid 2189 | . . . 4 ⊢ ∅ = ∅ | |
14 | 13 | orci 732 | . . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅) |
15 | peano3 4610 | . . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅) | |
16 | 15 | neneqd 2381 | . . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅) |
17 | 16 | olcd 735 | . . . 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 4614 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) |
20 | df-dc 836 | . 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) | |
21 | 19, 20 | sylibr 134 | 1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2160 ∅c0 3437 suc csuc 4380 ωcom 4604 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4386 df-iom 4605 |
This theorem is referenced by: omp1eomlem 7111 ctmlemr 7125 nnnninfeq2 7145 nninfisol 7149 elni2 7331 indpi 7359 nnsf 15139 peano4nninf 15140 |
Copyright terms: Public domain | W3C validator |