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Mirrors > Home > ILE Home > Th. List > ltdcpi | GIF version |
Description: Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
ltdcpi | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7058 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7058 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nndcel 6347 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵) | |
4 | 1, 2, 3 | syl2an 285 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 ∈ 𝐵) |
5 | ltpiord 7068 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | 5 | dcbid 806 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (DECID 𝐴 <N 𝐵 ↔ DECID 𝐴 ∈ 𝐵)) |
7 | 4, 6 | mpbird 166 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 802 ∈ wcel 1461 class class class wbr 3893 ωcom 4462 Ncnpi 7021 <N clti 7024 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-tr 3985 df-eprel 4169 df-iord 4246 df-on 4248 df-suc 4251 df-iom 4463 df-xp 4503 df-ni 7053 df-lti 7056 |
This theorem is referenced by: ltdcnq 7146 |
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