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Mirrors > Home > ILE Home > Th. List > eluzelz | GIF version |
Description: A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
eluzelz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 9325 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp2bi 997 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 ≤ cle 7794 ℤcz 9047 ℤ≥cuz 9319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-neg 7929 df-z 9048 df-uz 9320 |
This theorem is referenced by: eluzelre 9329 uztrn 9335 uzneg 9337 uzssz 9338 uzss 9339 eluzp1l 9343 eluzaddi 9345 eluzsubi 9346 eluzadd 9347 eluzsub 9348 uzm1 9349 uzin 9351 uzind4 9376 uz2mulcl 9395 elfz5 9791 elfzel2 9797 elfzelz 9799 eluzfz2 9805 peano2fzr 9810 fzsplit2 9823 fzopth 9834 fzsuc 9842 elfzp1 9845 fzdifsuc 9854 uzsplit 9865 uzdisj 9866 fzm1 9873 fzneuz 9874 uznfz 9876 nn0disj 9908 elfzo3 9933 fzoss2 9942 fzouzsplit 9949 eluzgtdifelfzo 9967 fzosplitsnm1 9979 fzofzp1b 9998 elfzonelfzo 10000 fzosplitsn 10003 fzisfzounsn 10006 mulp1mod1 10131 m1modge3gt1 10137 frec2uzltd 10169 frecfzen2 10193 uzennn 10202 uzsinds 10208 seq3fveq2 10235 seq3feq2 10236 seq3shft2 10239 monoord 10242 monoord2 10243 ser3mono 10244 seq3split 10245 iseqf1olemjpcl 10261 iseqf1olemqpcl 10262 seq3f1olemqsumk 10265 seq3f1olemp 10268 seq3f1oleml 10269 seq3f1o 10270 seq3id 10274 seq3z 10277 fser0const 10282 leexp2a 10339 expnlbnd2 10410 hashfz 10560 hashfzo 10561 hashfzp1 10563 seq3coll 10578 seq3shft 10603 rexuz3 10755 r19.2uz 10758 cau4 10881 caubnd2 10882 clim 11043 climshft2 11068 climaddc1 11091 climmulc2 11093 climsubc1 11094 climsubc2 11095 clim2ser 11099 clim2ser2 11100 iserex 11101 climlec2 11103 climub 11106 climcau 11109 climcaucn 11113 serf0 11114 sumrbdclem 11138 fsum3cvg 11139 summodclem2a 11143 zsumdc 11146 fsum3 11149 fisumss 11154 fsum3cvg2 11156 fsum3ser 11159 fsumcl2lem 11160 fsumadd 11168 fsumm1 11178 fzosump1 11179 fsum1p 11180 fsump1 11182 fsummulc2 11210 telfsumo 11228 fsumparts 11232 iserabs 11237 binomlem 11245 isumshft 11252 isumsplit 11253 isumrpcl 11256 divcnv 11259 trireciplem 11262 geosergap 11268 geolim2 11274 cvgratnnlemseq 11288 cvgratnnlemabsle 11289 cvgratnnlemsumlt 11290 cvgratnnlemrate 11292 cvgratz 11294 cvgratgt0 11295 mertenslemi1 11297 clim2divap 11302 prodrbdclem 11333 fproddccvg 11334 efgt1p2 11390 zsupcllemstep 11627 infssuzex 11631 dvdsbnd 11634 ncoprmgcdne1b 11759 isprm3 11788 prmind2 11790 nprm 11793 dvdsprm 11806 exprmfct 11807 phibndlem 11881 phibnd 11882 dfphi2 11885 hashdvds 11886 supfz 13226 |
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