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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 9644 |
. 2
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2 | eluzle 9188 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-neg 7807 df-z 8907 df-uz 9177 df-fz 9632 |
This theorem is referenced by: elfz1eq 9656 fzdisj 9673 fznatpl1 9697 fzp1disj 9701 uzdisj 9714 fzneuz 9722 fznuz 9723 elfzmlbm 9749 difelfznle 9753 nn0disj 9756 iseqf1olemqcl 10100 iseqf1olemnab 10102 iseqf1olemab 10103 iseqf1olemqk 10108 iseqf1olemfvp 10111 seq3f1olemqsumkj 10112 seq3f1olemqsumk 10113 seq3f1olemqsum 10114 seq3f1oleml 10117 seq3f1o 10118 bcval4 10339 bcp1nk 10349 zfz1isolemiso 10423 seq3coll 10426 summodclem3 10988 summodclem2a 10989 fsum3 10995 fsumcl2lem 11006 fsum0diaglem 11048 mertenslemi1 11143 fzm1ndvds 11349 prmind2 11594 prmdvdsfz 11612 hashdvds 11689 ennnfonelemim 11729 ctinfomlemom 11732 supfz 12821 |
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