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Mirrors > Home > MPE Home > Th. List > infmssuzcl | Structured version Unicode version |
Description: The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) |
Ref | Expression |
---|---|
infmssuzcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzssz 10986 |
. . . . 5
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2 | zssre 10759 |
. . . . 5
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3 | 1, 2 | sstri 3468 |
. . . 4
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4 | sstr 3467 |
. . . 4
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5 | 3, 4 | mpan2 671 |
. . 3
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6 | 5 | adantr 465 |
. 2
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7 | uzwo 11023 |
. 2
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8 | lbinfmcl 10390 |
. 2
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9 | 6, 7, 8 | syl2anc 661 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 ax-cnex 9444 ax-resscn 9445 ax-1cn 9446 ax-icn 9447 ax-addcl 9448 ax-addrcl 9449 ax-mulcl 9450 ax-mulrcl 9451 ax-mulcom 9452 ax-addass 9453 ax-mulass 9454 ax-distr 9455 ax-i2m1 9456 ax-1ne0 9457 ax-1rid 9458 ax-rnegex 9459 ax-rrecex 9460 ax-cnre 9461 ax-pre-lttri 9462 ax-pre-lttrn 9463 ax-pre-ltadd 9464 ax-pre-mulgt0 9465 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-nel 2648 df-ral 2801 df-rex 2802 df-reu 2803 df-rmo 2804 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-iun 4276 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-riota 6156 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-om 6582 df-recs 6937 df-rdg 6971 df-er 7206 df-en 7416 df-dom 7417 df-sdom 7418 df-sup 7797 df-pnf 9526 df-mnf 9527 df-xr 9528 df-ltxr 9529 df-le 9530 df-sub 9703 df-neg 9704 df-nn 10429 df-n0 10686 df-z 10753 df-uz 10968 |
This theorem is referenced by: zsupss 11050 uzwo3 11054 divalglem2 13712 bitsfzolem 13743 bezoutlem2 13836 odzcllem 13977 4sqlem13 14131 4sqlem14 14132 4sqlem17 14135 4sqlem18 14136 vdwnnlem3 14171 ramcl2lem 14183 ramtcl 14184 odlem1 16154 odlem2 16158 gexlem1 16194 gexlem2 16197 zringlpirlem2 18024 zringlpirlem3 18025 zlpirlem2 18029 zlpirlem3 18030 ovolicc2lem4 21130 iundisj 21157 ig1peu 21771 ig1pdvds 21776 elqaalem1 21913 elqaalem3 21915 ftalem4 22541 ftalem5 22542 iundisjf 26077 iundisjfi 26220 dgraalem 29645 |
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