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Mirrors > Home > ILE Home > Th. List > uztrn2 | Unicode version |
Description: Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.) |
Ref | Expression |
---|---|
uztrn2.1 |
Ref | Expression |
---|---|
uztrn2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uztrn2.1 | . . . 4 | |
2 | 1 | eleq2i 2206 | . . 3 |
3 | uztrn 9342 | . . . 4 | |
4 | 3 | ancoms 266 | . . 3 |
5 | 2, 4 | sylanb 282 | . 2 |
6 | 5, 1 | eleqtrrdi 2233 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cfv 5123 cuz 9326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltwlin 7733 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-neg 7936 df-z 9055 df-uz 9327 |
This theorem is referenced by: eluznn0 9393 eluznn 9394 elfzuz2 9809 rexuz3 10762 r19.29uz 10764 r19.2uz 10765 clim2 11052 clim2c 11053 clim0c 11055 2clim 11070 climabs0 11076 climcn1 11077 climcn2 11078 climsqz 11104 climsqz2 11105 clim2ser 11106 clim2ser2 11107 climub 11113 serf0 11121 mertenslemi1 11304 clim2divap 11309 lmbrf 12384 lmss 12415 lmres 12417 txlm 12448 |
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