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| Mirrors > Home > MPE Home > Th. List > eluzel2 | Structured version Visualization version GIF version | ||
| Description: Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
| Ref | Expression |
|---|---|
| eluzel2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvdm 6943 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ dom ℤ≥) | |
| 2 | uzf 12881 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
| 3 | 2 | fdmi 6747 | . 2 ⊢ dom ℤ≥ = ℤ |
| 4 | 1, 3 | eleqtrdi 2851 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: eluz2 12884 uztrn 12896 uzneg 12898 uzss 12901 uz11 12903 eluzadd 12907 eluzaddOLD 12913 subeluzsub 12915 uzm1 12916 uzin 12918 uzind4 12948 uzsupss 12982 elfz5 13556 elfzel1 13563 eluzfz1 13571 fzsplit2 13589 fzopth 13601 ssfzunsn 13610 fzpred 13612 fzpreddisj 13613 uzsplit 13636 uzdisj 13637 fzdif1 13645 fzm1 13647 uznfz 13650 nn0disj 13684 preduz 13690 fzolb 13705 fzoss2 13727 fzouzdisj 13735 fzoun 13736 ige2m2fzo 13767 fzen2 14010 seqp1 14057 seqcl 14063 seqfeq2 14066 seqfveq 14067 seqshft2 14069 seqsplit 14076 seqcaopr3 14078 seqf1olem2a 14081 seqf1olem1 14082 seqf1olem2 14083 seqid 14088 seqhomo 14090 seqz 14091 leexp2a 14212 hashfz 14466 fzsdom2 14467 hashfzo 14468 hashfzp1 14470 seqcoll 14503 rexanuz2 15388 cau4 15395 clim2ser 15691 clim2ser2 15692 climserle 15699 caurcvg 15713 caucvg 15715 fsumcvg 15748 fsumcvg2 15763 fsumsers 15764 fsumm1 15787 fsum1p 15789 fsumrev2 15818 telfsumo 15838 fsumparts 15842 cvgcmp 15852 cvgcmpub 15853 cvgcmpce 15854 isumsplit 15876 clim2prod 15924 clim2div 15925 prodfrec 15931 ntrivcvgtail 15936 fprodcvg 15966 fprodser 15985 fprodm1 16003 fprodeq0 16011 pcaddlem 16926 vdwnnlem2 17034 prmlem0 17143 gsumval2a 18698 telgsumfzs 20007 dvfsumle 26060 dvfsumleOLD 26061 dvfsumge 26062 dvfsumabs 26063 coeid3 26279 ulmres 26431 ulmss 26440 chtdif 27201 ppidif 27206 bcmono 27321 axlowdimlem6 28962 inffz 35730 mettrifi 37764 jm2.25 43011 jm2.16nn0 43016 dvgrat 44331 ssinc 45092 ssdec 45093 fzdifsuc2 45322 iuneqfzuzlem 45345 ssuzfz 45360 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 carageniuncllem1 46536 caratheodorylem1 46541 |
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