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| Mirrors > Home > MPE Home > Th. List > eluz | Structured version Visualization version GIF version | ||
| Description: Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
| Ref | Expression |
|---|---|
| eluz | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eluz1 12882 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
| 2 | 1 | baibd 539 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: uzneg 12898 uztric 12902 uzwo3 12985 fzn 13580 fzsplit2 13589 fznn 13632 uzsplit 13636 elfz2nn0 13658 fzouzsplit 13734 faclbnd 14329 bcval5 14357 fz1isolem 14500 seqcoll 14503 rexuzre 15391 caurcvg 15713 caucvg 15715 summolem2a 15751 fsum0diaglem 15812 climcnds 15887 mertenslem1 15920 ntrivcvgmullem 15937 prodmolem2a 15970 ruclem10 16275 eulerthlem2 16819 pcpremul 16881 pcdvdsb 16907 pcadd 16927 pcfac 16937 pcbc 16938 prmunb 16952 prmreclem5 16958 vdwnnlem3 17035 lt6abl 19913 ovolunlem1a 25531 mbflimsup 25701 plyco0 26231 plyeq0lem 26249 aannenlem1 26370 aaliou3lem2 26385 aaliou3lem8 26387 chtublem 27255 bcmax 27322 bpos1lem 27326 bposlem1 27328 axlowdimlem16 28972 fzsplit3 32795 cycpmco2lem7 33152 ballotlem2 34491 ballotlemimin 34508 breprexplemc 34647 elfzm12 35680 poimirlem3 37630 poimirlem4 37631 poimirlem28 37655 mblfinlem2 37665 incsequz 37755 incsequz2 37756 aks4d1p1 42077 primrootspoweq0 42107 aks6d1c2 42131 sticksstones12a 42158 sticksstones12 42159 aks6d1c6lem3 42173 nacsfix 42723 ellz1 42778 eluzrabdioph 42817 monotuz 42953 expdiophlem1 43033 nznngen 44335 fzisoeu 45312 fmul01 45595 climsuselem1 45622 climsuse 45623 iblspltprt 45988 itgspltprt 45994 wallispilem5 46084 stirlinglem8 46096 dirkertrigeqlem1 46113 fourierdlem12 46134 ssfz12 47326 |
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