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Mirrors > Home > ILE Home > Th. List > elfz2 | Unicode version |
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 398 | . 2 | |
2 | df-3an 949 | . . 3 | |
3 | 2 | anbi1i 453 | . 2 |
4 | df-fz 9759 | . . . 4 | |
5 | 4 | elmpocl 5936 | . . 3 |
6 | simpl 108 | . . 3 | |
7 | elfz1 9763 | . . . 4 | |
8 | 3anass 951 | . . . . 5 | |
9 | ibar 299 | . . . . 5 | |
10 | 8, 9 | syl5bb 191 | . . . 4 |
11 | 7, 10 | bitrd 187 | . . 3 |
12 | 5, 6, 11 | pm5.21nii 678 | . 2 |
13 | 1, 3, 12 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 947 wcel 1465 crab 2397 class class class wbr 3899 (class class class)co 5742 cle 7769 cz 9022 cfz 9758 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-neg 7904 df-z 9023 df-fz 9759 |
This theorem is referenced by: elfz4 9767 elfzuzb 9768 uzsubsubfz 9795 fzmmmeqm 9806 fzpreddisj 9819 elfz1b 9838 fzp1nel 9852 elfz0ubfz0 9870 elfz0fzfz0 9871 fz0fzelfz0 9872 fz0fzdiffz0 9875 elfzmlbp 9877 fzind2 9984 iseqf1olemqcl 10227 iseqf1olemnab 10229 iseqf1olemab 10230 seq3f1olemqsumkj 10239 seq3f1olemqsumk 10240 summodclem2a 11118 fsum3 11124 fsum3cvg3 11133 fsumcl2lem 11135 fsumadd 11143 fsummulc2 11185 |
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