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Mirrors > Home > MPE Home > Th. List > elfz | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.) |
Ref | Expression |
---|---|
elfz | ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz1 12900 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
2 | 3anass 1091 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) ↔ (𝐾 ∈ ℤ ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
3 | 2 | baib 538 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
4 | 1, 3 | sylan9bb 512 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
5 | 4 | 3impa 1106 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
6 | 5 | 3comr 1121 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 class class class wbr 5069 (class class class)co 7159 ≤ cle 10679 ℤcz 11984 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-neg 10876 df-z 11985 df-fz 12896 |
This theorem is referenced by: elfz5 12903 fzadd2 12945 fznatpl1 12964 fzrev 12973 fzctr 13022 elfzo 13043 seqf1olem1 13412 bcval5 13681 pfxccat3a 14103 isprm3 16030 hashdvds 16115 eulerthlem2 16122 prmreclem5 16259 aannenlem1 24920 basellem3 25663 chtub 25791 bcmono 25856 bposlem1 25863 lgseisenlem1 25954 lgsquadlem1 25959 2lgslem1a 25970 axlowdimlem3 26733 axlowdimlem7 26737 axlowdimlem16 26746 axlowdimlem17 26747 axlowdim 26750 submateqlem1 31076 lmatfvlem 31084 bcneg1 32972 poimirlem15 34911 poimirlem24 34920 poimirlem28 34924 mblfinlem2 34934 itg2addnclem2 34948 fzmul 35020 cntotbnd 35078 fzsplit1nn0 39357 irrapxlem3 39427 pellexlem5 39436 acongrep 39583 fzneg 39585 jm2.23 39599 fmul01 41867 fmuldfeq 41870 stoweidlem26 42318 fourierdlem11 42410 fourierdlem12 42411 fourierdlem15 42414 fourierdlem79 42477 smfmullem4 43076 |
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