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Mirrors > Home > ILE Home > Th. List > zsupcl | Unicode version |
Description: Closure of supremum for
decidable integer properties. The property
which defines the set we are taking the supremum of must (a) be true at
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Ref | Expression |
---|---|
zsupcl.m |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
zsupcl.sbm |
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zsupcl.mtru |
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zsupcl.dc |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
zsupcl.bnd |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
zsupcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsupcl.m |
. . . 4
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2 | 1 | zred 8778 |
. . 3
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3 | lttri3 7486 |
. . . . 5
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4 | 3 | adantl 271 |
. . . 4
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5 | zssre 8667 |
. . . . 5
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6 | zsupcl.sbm |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | zsupcl.mtru |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() | |
8 | zsupcl.dc |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | zsupcl.bnd |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 1, 6, 7, 8, 9 | zsupcllemex 10736 |
. . . . 5
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11 | ssrexv 3072 |
. . . . 5
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12 | 5, 10, 11 | mpsyl 64 |
. . . 4
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13 | 4, 12 | supclti 6614 |
. . 3
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14 | 6 | elrab 2761 |
. . . . 5
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15 | 1, 7, 14 | sylanbrc 408 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 4, 12 | supubti 6615 |
. . . 4
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17 | 15, 16 | mpd 13 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 2, 13, 17 | nltled 7525 |
. 2
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19 | 5 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 4, 10, 19 | supelti 6618 |
. . 3
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21 | eluz 8941 |
. . 3
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22 | 1, 20, 21 | syl2anc 403 |
. 2
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23 | 18, 22 | mpbird 165 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-sup 6600 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 df-fz 9334 df-fzo 9458 |
This theorem is referenced by: gcdsupcl 10744 |
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