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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 2046 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | 1 | notbid 592 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 = ∅ ↔ ¬ ∅ = ∅)) |
3 | 1, 2 | orbi12d 707 | . . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (∅ = ∅ ∨ ¬ ∅ = ∅))) |
4 | eqeq1 2046 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | 4 | notbid 592 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ 𝑦 = ∅)) |
6 | 4, 5 | orbi12d 707 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))) |
7 | eqeq1 2046 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | 7 | notbid 592 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 = ∅ ↔ ¬ suc 𝑦 = ∅)) |
9 | 7, 8 | orbi12d 707 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (suc 𝑦 = ∅ ∨ ¬ suc 𝑦 = ∅))) |
10 | eqeq1 2046 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | 10 | notbid 592 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 = ∅ ↔ ¬ 𝐴 = ∅)) |
12 | 10, 11 | orbi12d 707 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ¬ 𝑥 = ∅) ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅))) |
13 | eqid 2040 | . . . 4 ⊢ ∅ = ∅ | |
14 | 13 | orci 650 | . . 3 ⊢ (∅ = ∅ ∨ ¬ ∅ = ∅) |
15 | peano3 4319 | . . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ≠ ∅) | |
16 | 15 | neneqd 2226 | . . . . 5 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 = ∅) |
17 | 16 | olcd 653 | . . . 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 4323 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) |
20 | df-dc 743 | . 2 ⊢ (DECID 𝐴 = ∅ ↔ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)) | |
21 | 19, 20 | sylibr 137 | 1 ⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 DECID wdc 742 = wceq 1243 ∈ wcel 1393 ∅c0 3224 suc csuc 4102 ωcom 4313 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 |
This theorem is referenced by: elni2 6412 indpi 6440 |
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