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Mirrors > Home > ILE Home > Th. List > elfzle1 | GIF version |
Description: A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzle1 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 9426 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzle 9021 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 class class class wbr 3843 ‘cfv 5010 (class class class)co 5644 ≤ cle 7513 ℤ≥cuz 9009 ...cfz 9414 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-mpt 3899 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-fv 5018 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-neg 7646 df-z 8741 df-uz 9010 df-fz 9415 |
This theorem is referenced by: elfz1eq 9439 fzdisj 9456 elfznn 9458 fznatpl1 9478 fznn0sub2 9527 fz0fzdiffz0 9529 difelfznle 9534 iseqf1olemqcl 9903 iseqf1olemnab 9905 iseqf1olemab 9906 seq3f1olemqsumkj 9915 seq3f1olemqsumk 9916 seq3f1olemqsum 9917 bcval4 10148 iseqcoll 10235 fsum0diaglem 10821 cvgratnnlemabsle 10908 cvgratnnlemrate 10911 mertenslemi1 10916 hashdvds 11462 inffz 11800 |
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