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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 6407 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 6407 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nndcel 6078 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵) | |
4 | 1, 2, 3 | syl2an 273 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 ∈ 𝐵) |
5 | ltpiord 6417 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | 5 | dcbid 748 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (DECID 𝐴 <N 𝐵 ↔ DECID 𝐴 ∈ 𝐵)) |
7 | 4, 6 | mpbird 156 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 DECID wdc 742 ∈ wcel 1393 class class class wbr 3764 ωcom 4313 Ncnpi 6370 <N clti 6373 |
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-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 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-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-eprel 4026 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-ni 6402 df-lti 6405 |
This theorem is referenced by: ltdcnq 6495 |
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