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Mirrors > Home > ILE Home > Th. List > uztrn | 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 8478 | . . 3 | |
2 | 1 | adantl 262 | . 2 |
3 | eluzelz 8482 | . . 3 | |
4 | 3 | adantr 261 | . 2 |
5 | eluzle 8485 | . . . 4 | |
6 | 5 | adantl 262 | . . 3 |
7 | eluzle 8485 | . . . 4 | |
8 | 7 | adantr 261 | . . 3 |
9 | eluzelz 8482 | . . . . 5 | |
10 | 9 | adantl 262 | . . . 4 |
11 | zletr 8294 | . . . 4 | |
12 | 2, 10, 4, 11 | syl3anc 1135 | . . 3 |
13 | 6, 8, 12 | mp2and 409 | . 2 |
14 | eluz2 8479 | . 2 | |
15 | 2, 4, 13, 14 | syl3anbrc 1088 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wcel 1393 class class class wbr 3764 cfv 4902 cle 7061 cz 8245 cuz 8473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltwlin 6997 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-neg 7185 df-z 8246 df-uz 8474 |
This theorem is referenced by: uztrn2 8490 fzsplit2 8914 fzass4 8925 fzss1 8926 fzss2 8927 uzsplit 8954 iseqfveq2 9228 isermono 9237 iseqsplit 9238 iseqid 9247 |
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