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Mirrors > Home > ILE Home > Th. List > eluz2 | Unicode version |
Description: Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
eluz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 8624 | . 2 | |
2 | simp1 938 | . 2 | |
3 | eluz1 8623 | . . . 4 | |
4 | ibar 295 | . . . 4 | |
5 | 3, 4 | bitrd 186 | . . 3 |
6 | 3anass 923 | . . 3 | |
7 | 5, 6 | syl6bbr 196 | . 2 |
8 | 1, 2, 7 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 cfv 4922 cle 7154 cz 8351 cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-neg 7282 df-z 8352 df-uz 8620 |
This theorem is referenced by: eluzuzle 8627 eluzelz 8628 eluzle 8631 uztrn 8635 eluzp1p1 8644 uznn0sub 8650 uz3m2nn 8661 1eluzge0 8662 2eluzge1 8664 raluz2 8667 rexuz2 8669 peano2uz 8671 nn0pzuz 8675 uzind4 8676 nn0ge2m1nnALT 8703 elfzuzb 9039 uzsubsubfz 9066 ige2m1fz 9127 4fvwrd4 9150 elfzo2 9160 elfzouz2 9170 fzossrbm1 9182 fzossfzop1 9221 ssfzo12bi 9234 elfzonelfzo 9239 elfzomelpfzo 9240 fzosplitprm1 9243 fzostep1 9246 fzind2 9248 flqword2 9291 fldiv4p1lem1div2 9307 ibcval5 9690 resqrexlemoverl 9907 resqrexlemga 9909 oddge22np1 10281 nn0o 10307 dvdsnprmd 10507 prmgt1 10513 oddprmgt2 10515 oddprmge3 10516 |
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