![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 10053 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzle 9571 | . 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 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ≤ cle 8024 ℤ≥cuz 9559 ...cfz 10040 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-neg 8162 df-z 9285 df-uz 9560 df-fz 10041 |
This theorem is referenced by: elfz1eq 10067 fzdisj 10084 elfznn 10086 fznatpl1 10108 fznn0sub2 10160 fz0fzdiffz0 10162 difelfznle 10167 iseqf1olemqcl 10519 iseqf1olemnab 10521 iseqf1olemab 10522 seq3f1olemqsumkj 10531 seq3f1olemqsumk 10532 seq3f1olemqsum 10533 seqfeq4g 10546 bcval4 10767 seq3coll 10857 fsum0diaglem 11483 cvgratnnlemabsle 11570 cvgratnnlemrate 11573 mertenslemi1 11578 fprodntrivap 11627 prmdc 12165 hashdvds 12256 prmdiveq 12271 4sqlem11 12436 4sqlem12 12437 lgsdilem2 14915 inffz 15299 |
Copyright terms: Public domain | W3C validator |