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Mirrors > Home > ILE Home > Th. List > eluzdc | GIF version |
Description: Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.) |
Ref | Expression |
---|---|
eluzdc | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlelttric 9257 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ∨ 𝑁 < 𝑀)) | |
2 | eluz 9500 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | |
3 | 2 | biimprd 157 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 → 𝑁 ∈ (ℤ≥‘𝑀))) |
4 | zltnle 9258 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) | |
5 | 4 | ancoms 266 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) |
6 | 2 | notbid 662 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑁 ∈ (ℤ≥‘𝑀) ↔ ¬ 𝑀 ≤ 𝑁)) |
7 | 6 | biimprd 157 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 ≤ 𝑁 → ¬ 𝑁 ∈ (ℤ≥‘𝑀))) |
8 | 5, 7 | sylbid 149 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → ¬ 𝑁 ∈ (ℤ≥‘𝑀))) |
9 | 3, 8 | orim12d 781 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ≤ 𝑁 ∨ 𝑁 < 𝑀) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑁 ∈ (ℤ≥‘𝑀)))) |
10 | df-dc 830 | . . 3 ⊢ (DECID 𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑁 ∈ (ℤ≥‘𝑀))) | |
11 | 9, 10 | syl6ibr 161 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ≤ 𝑁 ∨ 𝑁 < 𝑀) → DECID 𝑁 ∈ (ℤ≥‘𝑀))) |
12 | 1, 11 | mpd 13 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 DECID wdc 829 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 < clt 7954 ≤ cle 7955 ℤcz 9212 ℤ≥cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: elnn0dc 9570 elnndc 9571 fzneuz 10057 sumdc 11321 summodclem2a 11344 zsumdc 11347 zproddc 11542 nninfdclemcl 12403 nninfdclemp1 12405 |
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