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Mirrors > Home > ILE Home > Th. List > eluzdc | Unicode version |
Description: Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.) |
Ref | Expression |
---|---|
eluzdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlelttric 8466 |
. 2
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2 | eluz 8702 |
. . . . 5
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3 | 2 | biimprd 156 |
. . . 4
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4 | zltnle 8467 |
. . . . . 6
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5 | 4 | ancoms 264 |
. . . . 5
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6 | 2 | notbid 625 |
. . . . . 6
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7 | 6 | biimprd 156 |
. . . . 5
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8 | 5, 7 | sylbid 148 |
. . . 4
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9 | 3, 8 | orim12d 733 |
. . 3
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10 | df-dc 777 |
. . 3
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11 | 9, 10 | syl6ibr 160 |
. 2
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12 | 1, 11 | mpd 13 |
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-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3898 ax-pow 3950 ax-pr 3966 ax-un 4190 ax-setind 4282 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-1re 7121 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-addcom 7127 ax-addass 7129 ax-distr 7131 ax-i2m1 7132 ax-0lt1 7133 ax-0id 7135 ax-rnegex 7136 ax-cnre 7138 ax-pre-ltirr 7139 ax-pre-ltwlin 7140 ax-pre-lttrn 7141 ax-pre-ltadd 7143 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-int 3639 df-br 3788 df-opab 3842 df-mpt 3843 df-id 4050 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-iota 4891 df-fun 4928 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-pnf 7206 df-mnf 7207 df-xr 7208 df-ltxr 7209 df-le 7210 df-sub 7337 df-neg 7338 df-inn 8096 df-n0 8345 df-z 8422 df-uz 8690 |
This theorem is referenced by: fzneuz 9183 sumdc 10322 |
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