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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 9497 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzle 9092 | . 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 1439 class class class wbr 3851 ‘cfv 5028 (class class class)co 5666 ≤ cle 7584 ℤ≥cuz 9080 ...cfz 9485 |
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-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-neg 7717 df-z 8812 df-uz 9081 df-fz 9486 |
This theorem is referenced by: elfz1eq 9510 fzdisj 9527 elfznn 9529 fznatpl1 9551 fznn0sub2 9600 fz0fzdiffz0 9602 difelfznle 9607 iseqf1olemqcl 9976 iseqf1olemnab 9978 iseqf1olemab 9979 seq3f1olemqsumkj 9988 seq3f1olemqsumk 9989 seq3f1olemqsum 9990 bcval4 10221 iseqcoll 10308 fsum0diaglem 10895 cvgratnnlemabsle 10982 cvgratnnlemrate 10985 mertenslemi1 10990 hashdvds 11536 inffz 12189 |
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