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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 9536 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp3bi 1014 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 ≤ cle 7995 ℤcz 9255 ℤ≥cuz 9530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-cnex 7904 ax-resscn 7905 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-neg 8133 df-z 9256 df-uz 9531 |
This theorem is referenced by: uztrn 9546 uzneg 9548 uzss 9550 uz11 9552 eluzp1l 9554 uzm1 9560 uzin 9562 uzind4 9590 elfz5 10019 elfzle1 10029 elfzle2 10030 elfzle3 10032 uzsplit 10094 uzdisj 10095 uznfz 10105 elfz2nn0 10114 uzsubfz0 10131 nn0disj 10140 fzouzdisj 10182 elfzonelfzo 10232 mulp1mod1 10367 m1modge3gt1 10373 uzennn 10438 seq3split 10481 seq3f1olemqsumk 10501 seq3f1o 10506 seq3coll 10824 seq3shft 10849 cvg1nlemcau 10995 resqrexlemcvg 11030 resqrexlemga 11034 summodclem2a 11391 fsum3 11397 fsum3cvg3 11406 fsumadd 11416 sumsnf 11419 fsummulc2 11458 isumshft 11500 divcnv 11507 geolim2 11522 cvgratnnlemseq 11536 cvgratnnlemsumlt 11538 cvgratz 11542 mertenslemi1 11545 prodmodclem3 11585 prodmodclem2a 11586 fprodntrivap 11594 prodsnf 11602 fprodeq0 11627 efcllemp 11668 infssuzex 11952 suprzubdc 11955 dvdsbnd 11959 uzwodc 12040 ncoprmgcdne1b 12091 isprm5 12144 hashdvds 12223 pcmpt2 12344 pcfaclem 12349 pcfac 12350 nninfdclemp1 12453 strext 12566 lgslem1 14486 lgsdirprm 14520 cvgcmp2nlemabs 14865 |
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