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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 ![]() ![]() ![]() |
Ref | Expression |
---|---|
eluz2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9022 |
. 2
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2 | simp1 943 |
. 2
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3 | eluz1 9021 |
. . . 4
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4 | ibar 295 |
. . . 4
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5 | 3, 4 | bitrd 186 |
. . 3
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6 | 3anass 928 |
. . 3
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7 | 5, 6 | syl6bbr 196 |
. 2
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8 | 1, 2, 7 | pm5.21nii 655 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-cnex 7434 ax-resscn 7435 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-ov 5655 df-neg 7654 df-z 8749 df-uz 9018 |
This theorem is referenced by: eluzuzle 9025 eluzelz 9026 eluzle 9029 uztrn 9033 eluzp1p1 9042 uznn0sub 9048 uz3m2nn 9059 1eluzge0 9060 2eluzge1 9062 raluz2 9065 rexuz2 9067 peano2uz 9069 nn0pzuz 9073 uzind4 9074 nn0ge2m1nnALT 9101 elfzuzb 9432 uzsubsubfz 9459 ige2m1fz 9520 4fvwrd4 9547 elfzo2 9557 elfzouz2 9568 fzossrbm1 9580 fzossfzop1 9619 ssfzo12bi 9632 elfzonelfzo 9637 elfzomelpfzo 9638 fzosplitprm1 9641 fzostep1 9644 fzind2 9646 flqword2 9692 fldiv4p1lem1div2 9708 seq3split 9903 iseqf1olemqk 9919 seq3f1olemqsumkj 9923 seq3f1olemqsumk 9924 seq3f1olemqsum 9925 ibcval5 10167 iseqcoll 10243 seq3shft 10268 resqrexlemoverl 10450 resqrexlemga 10452 fsum3cvg3 10785 fisumrev2 10836 isumshft 10880 cvgratnnlemseq 10916 cvgratnnlemabsle 10917 cvgratnnlemsumlt 10918 cvgratz 10922 oddge22np1 11155 nn0o 11181 dvdsnprmd 11381 prmgt1 11387 oddprmgt2 11389 oddprmge3 11390 |
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