Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9299 | . 2 | |
2 | simp1 966 | . 2 | |
3 | eluz1 9298 | . . . 4 | |
4 | ibar 299 | . . . 4 | |
5 | 3, 4 | bitrd 187 | . . 3 |
6 | 3anass 951 | . . 3 | |
7 | 5, 6 | syl6bbr 197 | . 2 |
8 | 1, 2, 7 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 947 wcel 1465 class class class wbr 3899 cfv 5093 cle 7769 cz 9022 cuz 9294 |
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-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-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 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-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-neg 7904 df-z 9023 df-uz 9295 |
This theorem is referenced by: eluzuzle 9302 eluzelz 9303 eluzle 9306 uztrn 9310 eluzp1p1 9319 uznn0sub 9325 uz3m2nn 9336 1eluzge0 9337 2eluzge1 9339 raluz2 9342 rexuz2 9344 peano2uz 9346 nn0pzuz 9350 uzind4 9351 nn0ge2m1nnALT 9378 elfzuzb 9768 uzsubsubfz 9795 ige2m1fz 9858 4fvwrd4 9885 elfzo2 9895 elfzouz2 9906 fzossrbm1 9918 fzossfzop1 9957 ssfzo12bi 9970 elfzonelfzo 9975 elfzomelpfzo 9976 fzosplitprm1 9979 fzostep1 9982 fzind2 9984 flqword2 10030 fldiv4p1lem1div2 10046 uzennn 10177 seq3split 10220 iseqf1olemqk 10235 seq3f1olemqsumkj 10239 seq3f1olemqsumk 10240 seq3f1olemqsum 10241 bcval5 10477 seq3coll 10553 seq3shft 10578 resqrexlemoverl 10761 resqrexlemga 10763 fsum3cvg3 11133 fisumrev2 11183 isumshft 11227 cvgratnnlemseq 11263 cvgratnnlemabsle 11264 cvgratnnlemsumlt 11265 cvgratz 11269 oddge22np1 11505 nn0o 11531 dvdsnprmd 11733 prmgt1 11739 oddprmgt2 11741 oddprmge3 11742 strleund 11974 strleun 11975 |
Copyright terms: Public domain | W3C validator |