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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 9359 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
2 | eluzel2 9355 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | jca 304 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) |
4 | elfz 9827 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
5 | 4 | 3expa 1182 | . . 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 9362 | . . . 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 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ≤ cle 7825 ℤcz 9078 ℤ≥cuz 9350 ...cfz 9821 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-neg 7960 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: fzsplit2 9861 fznn0sub2 9936 iseqf1olemjpcl 10299 iseqf1olemqpcl 10300 seq3f1oleml 10307 bcval5 10541 seq3coll 10617 fsum0diaglem 11241 mertenslemi1 11336 |
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