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Mirrors > Home > ILE Home > Th. List > elfzle2 | Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz3 10040 |
. 2
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2 | eluzle 9558 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 4189 ax-pr 4224 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 |
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 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-neg 8149 df-z 9272 df-uz 9547 df-fz 10027 |
This theorem is referenced by: elfz1eq 10053 fzdisj 10070 fznatpl1 10094 fzp1disj 10098 uzdisj 10111 fzneuz 10119 fznuz 10120 elfzmlbm 10149 difelfznle 10153 nn0disj 10156 iseqf1olemqcl 10504 iseqf1olemnab 10506 iseqf1olemab 10507 iseqf1olemqk 10512 iseqf1olemfvp 10515 seq3f1olemqsumkj 10516 seq3f1olemqsumk 10517 seq3f1olemqsum 10518 seq3f1oleml 10521 seq3f1o 10522 bcval4 10750 bcp1nk 10760 zfz1isolemiso 10837 seq3coll 10840 summodclem3 11406 summodclem2a 11407 fsum3 11413 fsumcl2lem 11424 fsum0diaglem 11466 mertenslemi1 11561 prodmodclem3 11601 prodmodclem2a 11602 fprodseq 11609 fzm1ndvds 11880 prmind2 12138 prmdvdsfz 12157 isprm5lem 12159 hashdvds 12239 eulerthlemrprm 12247 eulerthlema 12248 prmdiveq 12254 ennnfonelemim 12443 ctinfomlemom 12446 lgsval2lem 14808 lgseisenlem1 14847 lgseisenlem2 14848 supfz 15217 |
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