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Mirrors > Home > ILE Home > Th. List > elfz5 | GIF version |
Description: Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.) |
Ref | Expression |
---|---|
elfz5 | ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9467 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
2 | eluzel2 9463 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | jca 304 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) |
4 | elfz 9942 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
5 | 4 | 3expa 1192 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
6 | 3, 5 | sylan 281 | . 2 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
7 | eluzle 9470 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾) | |
8 | 7 | biantrurd 303 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 ≤ 𝑁 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
9 | 8 | adantr 274 | . 2 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ≤ 𝑁 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
10 | 6, 9 | bitr4d 190 | 1 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 class class class wbr 3977 ‘cfv 5183 (class class class)co 5837 ≤ cle 7926 ℤcz 9183 ℤ≥cuz 9458 ...cfz 9936 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-neg 8064 df-z 9184 df-uz 9459 df-fz 9937 |
This theorem is referenced by: fzsplit2 9976 fznn0sub2 10054 iseqf1olemjpcl 10421 iseqf1olemqpcl 10422 seq3f1oleml 10429 bcval5 10666 seq3coll 10742 fsum0diaglem 11368 mertenslemi1 11463 fprodmul 11519 eulerthlemrprm 12147 eulerthlema 12148 pcfac 12266 |
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