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Mirrors > Home > ILE Home > Th. List > eluzle | GIF version |
Description: Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
eluzle | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 9227 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp3bi 979 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1461 class class class wbr 3893 ‘cfv 5079 ≤ cle 7718 ℤcz 8951 ℤ≥cuz 9221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-cnex 7629 ax-resscn 7630 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-ov 5729 df-neg 7852 df-z 8952 df-uz 9222 |
This theorem is referenced by: uztrn 9237 uzneg 9239 uzss 9241 uz11 9243 eluzp1l 9245 uzm1 9251 uzin 9253 uzind4 9278 elfz5 9684 elfzle1 9693 elfzle2 9694 elfzle3 9696 uzsplit 9758 uzdisj 9759 uznfz 9769 elfz2nn0 9778 uzsubfz0 9792 nn0disj 9801 fzouzdisj 9843 elfzonelfzo 9893 mulp1mod1 10024 m1modge3gt1 10030 uzennn 10095 seq3split 10138 seq3f1olemqsumk 10158 seq3f1o 10163 seq3coll 10471 seq3shft 10496 cvg1nlemcau 10641 resqrexlemcvg 10676 resqrexlemga 10680 summodclem2a 11035 fsum3 11041 fsum3cvg3 11050 fsumadd 11060 sumsnf 11063 fsummulc2 11102 isumshft 11144 divcnv 11151 geolim2 11166 cvgratnnlemseq 11180 cvgratnnlemsumlt 11182 cvgratz 11186 mertenslemi1 11189 efcllemp 11208 infssuzex 11483 dvdsbnd 11486 ncoprmgcdne1b 11609 hashdvds 11735 cvgcmp2nlemabs 12904 |
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