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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 6928 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ dom ℤ≥) | |
2 | uzf 12849 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
3 | 2 | fdmi 6728 | . 2 ⊢ dom ℤ≥ = ℤ |
4 | 1, 3 | eleqtrdi 2838 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 𝒫 cpw 4598 dom cdm 5672 ‘cfv 6542 ℤcz 12582 ℤ≥cuz 12846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-cnex 11188 ax-resscn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-neg 11471 df-z 12583 df-uz 12847 |
This theorem is referenced by: eluz2 12852 uztrn 12864 uzneg 12866 uzss 12869 uz11 12871 eluzadd 12875 eluzaddOLD 12881 subeluzsub 12883 uzm1 12884 uzin 12886 uzind4 12914 uzsupss 12948 elfz5 13519 elfzel1 13526 eluzfz1 13534 fzsplit2 13552 fzopth 13564 ssfzunsn 13573 fzpred 13575 fzpreddisj 13576 uzsplit 13599 uzdisj 13600 fzm1 13607 uznfz 13610 nn0disj 13643 preduz 13649 fzolb 13664 fzoss2 13686 fzouzdisj 13694 fzoun 13695 ige2m2fzo 13721 fzen2 13960 seqp1 14007 seqcl 14013 seqfeq2 14016 seqfveq 14017 seqshft2 14019 seqsplit 14026 seqcaopr3 14028 seqf1olem2a 14031 seqf1olem1 14032 seqf1olem2 14033 seqid 14038 seqhomo 14040 seqz 14041 leexp2a 14162 hashfz 14412 fzsdom2 14413 hashfzo 14414 hashfzp1 14416 seqcoll 14451 rexanuz2 15322 cau4 15329 clim2ser 15627 clim2ser2 15628 climserle 15635 caurcvg 15649 caucvg 15651 fsumcvg 15684 fsumcvg2 15699 fsumsers 15700 fsumm1 15723 fsum1p 15725 fsumrev2 15754 telfsumo 15774 fsumparts 15778 cvgcmp 15788 cvgcmpub 15789 cvgcmpce 15790 isumsplit 15812 clim2prod 15860 clim2div 15861 prodfrec 15867 ntrivcvgtail 15872 fprodcvg 15900 fprodser 15919 fprodm1 15937 fprodeq0 15945 pcaddlem 16850 vdwnnlem2 16958 prmlem0 17068 gsumval2a 18638 telgsumfzs 19937 dvfsumle 25947 dvfsumleOLD 25948 dvfsumge 25949 dvfsumabs 25950 coeid3 26167 ulmres 26317 ulmss 26326 chtdif 27083 ppidif 27088 bcmono 27203 axlowdimlem6 28751 inffz 35314 mettrifi 37219 jm2.25 42392 jm2.16nn0 42397 dvgrat 43721 ssinc 44425 ssdec 44426 fzdifsuc2 44664 iuneqfzuzlem 44688 ssuzfz 44703 ioodvbdlimc1lem2 45292 ioodvbdlimc2lem 45294 carageniuncllem1 45881 caratheodorylem1 45886 |
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