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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 9332 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp3bi 998 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 ≤ cle 7801 ℤcz 9054 ℤ≥cuz 9326 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-cnex 7711 ax-resscn 7712 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-neg 7936 df-z 9055 df-uz 9327 |
This theorem is referenced by: uztrn 9342 uzneg 9344 uzss 9346 uz11 9348 eluzp1l 9350 uzm1 9356 uzin 9358 uzind4 9383 elfz5 9798 elfzle1 9807 elfzle2 9808 elfzle3 9810 uzsplit 9872 uzdisj 9873 uznfz 9883 elfz2nn0 9892 uzsubfz0 9906 nn0disj 9915 fzouzdisj 9957 elfzonelfzo 10007 mulp1mod1 10138 m1modge3gt1 10144 uzennn 10209 seq3split 10252 seq3f1olemqsumk 10272 seq3f1o 10277 seq3coll 10585 seq3shft 10610 cvg1nlemcau 10756 resqrexlemcvg 10791 resqrexlemga 10795 summodclem2a 11150 fsum3 11156 fsum3cvg3 11165 fsumadd 11175 sumsnf 11178 fsummulc2 11217 isumshft 11259 divcnv 11266 geolim2 11281 cvgratnnlemseq 11295 cvgratnnlemsumlt 11297 cvgratz 11301 mertenslemi1 11304 prodmodclem3 11344 prodmodclem2a 11345 efcllemp 11364 infssuzex 11642 dvdsbnd 11645 ncoprmgcdne1b 11770 hashdvds 11897 cvgcmp2nlemabs 13227 |
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