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Mirrors > Home > MPE Home > Th. List > uztrn | Structured version Visualization version Unicode version |
Description: Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
uztrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 11692 | . . 3 | |
2 | 1 | adantl 482 | . 2 |
3 | eluzelz 11697 | . . 3 | |
4 | 3 | adantr 481 | . 2 |
5 | eluzle 11700 | . . . 4 | |
6 | 5 | adantl 482 | . . 3 |
7 | eluzle 11700 | . . . 4 | |
8 | 7 | adantr 481 | . . 3 |
9 | eluzelz 11697 | . . . . 5 | |
10 | 9 | adantl 482 | . . . 4 |
11 | zletr 11421 | . . . 4 | |
12 | 2, 10, 4, 11 | syl3anc 1326 | . . 3 |
13 | 6, 8, 12 | mp2and 715 | . 2 |
14 | eluz2 11693 | . 2 | |
15 | 2, 4, 13, 14 | syl3anbrc 1246 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 class class class wbr 4653 cfv 5888 cle 10075 cz 11377 cuz 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 |
This theorem is referenced by: uztrn2 11705 fzsplit2 12366 fzass4 12379 fzss1 12380 fzss2 12381 uzsplit 12412 seqfveq2 12823 sermono 12833 seqsplit 12834 seqid2 12847 fzsdom2 13215 seqcoll 13248 spllen 13505 splfv2a 13507 splval2 13508 climcndslem1 14581 mertenslem1 14616 ntrivcvgfvn0 14631 zprod 14667 dvdsfac 15048 smupvallem 15205 vdwlem2 15686 vdwlem6 15690 efgredleme 18156 bposlem6 25014 dchrisumlem2 25179 axlowdimlem16 25837 fzsplit3 29553 sseqf 30454 ballotlemsima 30577 ballotlemfrc 30588 climuzcnv 31565 seqpo 33543 incsequz2 33545 mettrifi 33553 monotuz 37506 |
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