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Mirrors > Home > ILE Home > Th. List > elfzel2 | 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 9796 | . 2 | |
2 | eluzelz 9328 | . 2 | |
3 | 1, 2 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 cfv 5118 (class class class)co 5767 cz 9047 cuz 9319 cfz 9783 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-neg 7929 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: elfz1eq 9808 fzdisj 9825 fzssp1 9840 fzp1disj 9853 fzrev2i 9859 fzrev3 9860 fznuz 9875 fznn0sub2 9898 elfzmlbm 9901 difelfznle 9905 nn0disj 9908 fzofzp1b 9998 iseqf1olemqcl 10252 iseqf1olemab 10255 iseqf1olemqf1o 10259 iseqf1olemqk 10260 iseqf1olemjpcl 10261 iseqf1olemqpcl 10262 iseqf1olemfvp 10263 seq3f1olemqsumkj 10264 seq3f1olemqsumk 10265 seq3f1olemqsum 10266 seq3f1olemstep 10267 bcm1k 10499 bcp1nk 10501 |
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