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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 2146 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | 1 | notbid 656 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅)) |
3 | 1, 2 | orbi12d 782 | . . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅))) |
4 | eqeq1 2146 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | 4 | notbid 656 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅)) |
6 | 4, 5 | orbi12d 782 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))) |
7 | eqeq1 2146 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | 7 | notbid 656 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅)) |
9 | 7, 8 | orbi12d 782 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))) |
10 | eqeq1 2146 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | 10 | notbid 656 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅)) |
12 | 10, 11 | orbi12d 782 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))) |
13 | eqid 2139 | . . . 4 ⊢ ∅ = ∅ | |
14 | 13 | orci 720 | . . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅) |
15 | peano3 4510 | . . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅) | |
16 | 15 | neneqd 2329 | . . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅) |
17 | 16 | olcd 723 | . . . 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 4514 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) |
20 | df-dc 820 | . 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) | |
21 | 19, 20 | sylibr 133 | 1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ∅c0 3363 suc csuc 4287 ωcom 4504 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: omp1eomlem 6979 ctmlemr 6993 elni2 7122 indpi 7150 nnsf 13199 peano4nninf 13200 |
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