Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8624 | . . 3 | |
2 | 1 | adantl 271 | . 2 |
3 | eluzelz 8628 | . . 3 | |
4 | 3 | adantr 270 | . 2 |
5 | eluzle 8631 | . . . 4 | |
6 | 5 | adantl 271 | . . 3 |
7 | eluzle 8631 | . . . 4 | |
8 | 7 | adantr 270 | . . 3 |
9 | eluzelz 8628 | . . . . 5 | |
10 | 9 | adantl 271 | . . . 4 |
11 | zletr 8400 | . . . 4 | |
12 | 2, 10, 4, 11 | syl3anc 1169 | . . 3 |
13 | 6, 8, 12 | mp2and 423 | . 2 |
14 | eluz2 8625 | . 2 | |
15 | 2, 4, 13, 14 | syl3anbrc 1122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wcel 1433 class class class wbr 3785 cfv 4922 cle 7154 cz 8351 cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltwlin 7089 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-neg 7282 df-z 8352 df-uz 8620 |
This theorem is referenced by: uztrn2 8636 fzsplit2 9069 fzass4 9080 fzss1 9081 fzss2 9082 uzsplit 9109 iseqfveq2 9448 isermono 9457 iseqsplit 9458 iseqid 9467 iseqz 9469 dvdsfac 10260 |
Copyright terms: Public domain | W3C validator |