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Mirrors > Home > ILE Home > Th. List > uzid | Unicode version |
Description: Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
uzid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9270 |
. . . 4
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2 | 1 | leidd 8484 |
. . 3
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3 | 2 | ancli 323 |
. 2
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4 | eluz1 9545 |
. 2
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5 | 3, 4 | mpbird 167 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-pre-ltirr 7936 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-ov 5891 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-neg 8144 df-z 9267 df-uz 9542 |
This theorem is referenced by: uzn0 9556 uz11 9563 eluzfz1 10044 eluzfz2 10045 elfz3 10047 elfz1end 10068 fzssp1 10080 fzpred 10083 fzp1ss 10086 fzpr 10090 fztp 10091 elfz0add 10133 fzolb 10166 zpnn0elfzo 10220 fzosplitsnm1 10222 fzofzp1 10240 fzosplitsn 10246 fzostep1 10250 frec2uzuzd 10415 frecuzrdgrrn 10421 frec2uzrdg 10422 frecuzrdgrcl 10423 frecuzrdgsuc 10427 frecuzrdgrclt 10428 frecuzrdgg 10429 frecuzrdgsuctlem 10436 uzsinds 10455 seq3val 10471 seqvalcd 10472 seq3-1 10473 seqf 10474 seq3p1 10475 seq3fveq 10484 seq3-1p 10493 seq3caopr3 10494 iseqf1olemjpcl 10508 iseqf1olemqpcl 10509 seq3f1oleml 10516 seq3f1o 10517 seq3homo 10523 faclbnd3 10736 bcm1k 10753 bcn2 10757 seq3coll 10835 rexuz3 11012 r19.2uz 11015 resqrexlemcvg 11041 resqrexlemgt0 11042 resqrexlemoverl 11043 cau3lem 11136 caubnd2 11139 climconst 11311 climuni 11314 climcau 11368 serf0 11373 fsumparts 11491 isum1p 11513 isumrpcl 11515 cvgratz 11553 mertenslemi1 11556 ntrivcvgap0 11570 fprodabs 11637 eftlub 11711 zsupcllemstep 11959 zsupcllemex 11960 ialgr0 12057 eucalg 12072 pw2dvds 12179 eulerthlemrprm 12242 oddprm 12272 pcfac 12361 pcbc 12362 ennnfonelem1 12421 lmconst 13987 2logb9irr 14660 sqrt2cxp2logb9e3 14664 2logb9irrap 14666 cvgcmp2nlemabs 15052 trilpolemlt1 15061 |
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