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Mirrors > Home > MPE Home > Th. List > elfz5 | Structured version Visualization version 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 12886 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
2 | eluzel2 12881 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | jca 511 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) |
4 | elfz 13550 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
5 | 4 | 3expa 1117 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
6 | 3, 5 | sylan 580 | . 2 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
7 | eluzle 12889 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾) | |
8 | 7 | biantrurd 532 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 ≤ 𝑁 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
9 | 8 | adantr 480 | . 2 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ≤ 𝑁 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
10 | 6, 9 | bitr4d 282 | 1 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ≤ cle 11294 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-neg 11493 df-z 12612 df-uz 12877 df-fz 13545 |
This theorem is referenced by: fzsplit2 13586 fznn0sub2 13672 predfz 13690 bcval5 14354 hashf1 14493 seqcoll 14500 limsupgre 15514 isercolllem2 15699 isercoll 15701 fsumcvg3 15762 fsum0diaglem 15809 climcndslem2 15883 mertenslem1 15917 ncoprmlnprm 16762 pcfac 16933 prmreclem2 16951 prmreclem3 16952 prmreclem5 16954 1arith 16961 vdwlem1 17015 vdwlem3 17017 vdwlem10 17024 sylow1lem1 19631 psrbaglefi 21964 ovoliunlem1 25551 ovolicc2lem4 25569 uniioombllem3 25634 mbfi1fseqlem3 25767 plyeq0lem 26264 coeeulem 26278 coeidlem 26291 coeid3 26294 coeeq2 26296 coemulhi 26308 vieta1lem2 26368 birthdaylem2 27010 birthdaylem3 27011 ftalem5 27135 basellem2 27140 basellem3 27141 basellem5 27143 musum 27249 fsumvma2 27273 chpchtsum 27278 lgsne0 27394 lgsquadlem2 27440 rplogsumlem2 27544 dchrisumlem1 27548 dchrisum0lem1 27575 ostth2lem3 27694 eupth2lems 30267 fzsplit3 32802 eulerpartlems 34342 eulerpartlemb 34350 erdszelem7 35182 cvmliftlem7 35276 |
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