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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 9463 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp3bi 1003 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 ≤ cle 7925 ℤcz 9182 ℤ≥cuz 9457 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-neg 8063 df-z 9183 df-uz 9458 |
This theorem is referenced by: uztrn 9473 uzneg 9475 uzss 9477 uz11 9479 eluzp1l 9481 uzm1 9487 uzin 9489 uzind4 9517 elfz5 9943 elfzle1 9952 elfzle2 9953 elfzle3 9955 uzsplit 10017 uzdisj 10018 uznfz 10028 elfz2nn0 10037 uzsubfz0 10054 nn0disj 10063 fzouzdisj 10105 elfzonelfzo 10155 mulp1mod1 10290 m1modge3gt1 10296 uzennn 10361 seq3split 10404 seq3f1olemqsumk 10424 seq3f1o 10429 seq3coll 10741 seq3shft 10766 cvg1nlemcau 10912 resqrexlemcvg 10947 resqrexlemga 10951 summodclem2a 11308 fsum3 11314 fsum3cvg3 11323 fsumadd 11333 sumsnf 11336 fsummulc2 11375 isumshft 11417 divcnv 11424 geolim2 11439 cvgratnnlemseq 11453 cvgratnnlemsumlt 11455 cvgratz 11459 mertenslemi1 11462 prodmodclem3 11502 prodmodclem2a 11503 fprodntrivap 11511 prodsnf 11519 fprodeq0 11544 efcllemp 11585 infssuzex 11867 suprzubdc 11870 dvdsbnd 11874 ncoprmgcdne1b 12000 isprm5 12053 hashdvds 12132 pcmpt2 12253 pcfaclem 12258 pcfac 12259 nninfdclemp1 12328 cvgcmp2nlemabs 13752 |
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