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Mirrors > Home > MPE Home > Th. List > elfzel2 | Structured version Visualization version Unicode version |
Description: Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzel2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz3 12339 | . 2 | |
2 | eluzelz 11697 | . 2 | |
3 | 1, 2 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 cfv 5888 (class class class)co 6650 cz 11377 cuz 11687 cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: elfz1eq 12352 fzdisj 12368 fzssp1 12384 fzp1disj 12399 fzrev2i 12405 fzrev3 12406 elfz1b 12409 fznuz 12422 fznn0sub2 12446 elfzmlbm 12449 difelfznle 12453 nn0disj 12455 fz1fzo0m1 12515 fzofzp1b 12566 bcm1k 13102 bcp1nk 13104 swrdccatin12lem2 13489 spllen 13505 fsum0diag2 14515 fallfacval3 14743 fallfacval4 14774 psgnunilem2 17915 pntpbnd1 25275 crctcshwlkn0 26713 elfzfzo 39488 sumnnodd 39862 dvnmul 40158 dvnprodlem1 40161 dvnprodlem2 40162 stoweidlem34 40251 fourierdlem11 40335 fourierdlem12 40336 fourierdlem15 40339 fourierdlem41 40365 fourierdlem48 40371 fourierdlem49 40372 fourierdlem54 40377 fourierdlem79 40402 fourierdlem102 40425 fourierdlem114 40437 etransclem23 40474 etransclem35 40486 iundjiun 40677 2elfz2melfz 41328 elfzelfzlble 41331 iccpartiltu 41358 iccpartgt 41363 pfxccatin12lem2 41424 |
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