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Mirrors > Home > ILE Home > Th. List > nndcel | GIF version |
Description: Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.) |
Ref | Expression |
---|---|
nndcel | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6254 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
2 | orc 668 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) | |
3 | elirr 4357 | . . . . . 6 ⊢ ¬ 𝐵 ∈ 𝐵 | |
4 | eleq1 2150 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐵 ∈ 𝐵)) | |
5 | 3, 4 | mtbiri 635 | . . . . 5 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
6 | 5 | olcd 688 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) |
7 | en2lp 4370 | . . . . . 6 ⊢ ¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) | |
8 | 7 | imnani 660 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐵) |
9 | 8 | olcd 688 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) |
10 | 2, 6, 9 | 3jaoi 1239 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) |
11 | 1, 10 | syl 14 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) |
12 | df-dc 781 | . 2 ⊢ (DECID 𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)) | |
13 | 11, 12 | sylibr 132 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 664 DECID wdc 780 ∨ w3o 923 = wceq 1289 ∈ wcel 1438 ωcom 4405 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-tr 3937 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 |
This theorem is referenced by: nnnninf 6804 ltdcpi 6880 nninfalllemn 11853 |
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