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Mirrors > Home > ILE Home > Th. List > elfzle2 | GIF version |
Description: A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzle2 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz3 9920 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | eluzle 9446 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝐾 ≤ 𝑁) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 class class class wbr 3965 ‘cfv 5169 (class class class)co 5821 ≤ cle 7908 ℤ≥cuz 9434 ...cfz 9907 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-neg 8044 df-z 9163 df-uz 9435 df-fz 9908 |
This theorem is referenced by: elfz1eq 9932 fzdisj 9949 fznatpl1 9973 fzp1disj 9977 uzdisj 9990 fzneuz 9998 fznuz 9999 elfzmlbm 10025 difelfznle 10029 nn0disj 10032 iseqf1olemqcl 10380 iseqf1olemnab 10382 iseqf1olemab 10383 iseqf1olemqk 10388 iseqf1olemfvp 10391 seq3f1olemqsumkj 10392 seq3f1olemqsumk 10393 seq3f1olemqsum 10394 seq3f1oleml 10397 seq3f1o 10398 bcval4 10621 bcp1nk 10631 zfz1isolemiso 10705 seq3coll 10708 summodclem3 11272 summodclem2a 11273 fsum3 11279 fsumcl2lem 11290 fsum0diaglem 11332 mertenslemi1 11427 prodmodclem3 11467 prodmodclem2a 11468 fprodseq 11475 fzm1ndvds 11742 prmind2 11991 prmdvdsfz 12010 hashdvds 12088 eulerthlemrprm 12096 eulerthlema 12097 prmdiveq 12103 ennnfonelemim 12140 ctinfomlemom 12143 supfz 13626 |
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