elfz5 - Intuitionistic Logic Explorer

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Theorem elfz5 9798

Description: Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)

**Assertion**
Ref	Expression
elfz5	⊢ ((𝐾 ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁))

**Proof of Theorem elfz5**
Step	Hyp	Ref	Expression
1		eluzelz 9335	. . . 4 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝐾 ∈ ℤ)
2		eluzel2 9331	. . . 4 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
3	1, 2	jca 304	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ))
4		elfz 9796	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
5	4	3expa 1181	. . 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 9338	. . . 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 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ≤ cle 7801 ℤcz 9054 ℤ_≥cuz 9326 ...cfz 9790

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-neg 7936 df-z 9055 df-uz 9327 df-fz 9791

This theorem is referenced by: fzsplit2 9830 fznn0sub2 9905 iseqf1olemjpcl 10268 iseqf1olemqpcl 10269 seq3f1oleml 10276 bcval5 10509 seq3coll 10585 fsum0diaglem 11209 mertenslemi1 11304

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