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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 9492 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | eluzelz 9496 | . . 3 | |
4 | 3 | adantr 274 | . 2 |
5 | eluzle 9499 | . . . 4 | |
6 | 5 | adantl 275 | . . 3 |
7 | eluzle 9499 | . . . 4 | |
8 | 7 | adantr 274 | . . 3 |
9 | eluzelz 9496 | . . . . 5 | |
10 | 9 | adantl 275 | . . . 4 |
11 | zletr 9261 | . . . 4 | |
12 | 2, 10, 4, 11 | syl3anc 1233 | . . 3 |
13 | 6, 8, 12 | mp2and 431 | . 2 |
14 | eluz2 9493 | . 2 | |
15 | 2, 4, 13, 14 | syl3anbrc 1176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 class class class wbr 3989 cfv 5198 cle 7955 cz 9212 cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltwlin 7887 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-neg 8093 df-z 9213 df-uz 9488 |
This theorem is referenced by: uztrn2 9504 fzsplit2 10006 fzass4 10018 fzss1 10019 fzss2 10020 uzsplit 10048 seq3fveq2 10425 ser3mono 10434 seq3split 10435 seq3f1olemqsumkj 10454 seq3f1olemqsumk 10455 seq3id 10464 seq3id2 10465 seq3z 10467 seq3coll 10777 cvgratgt0 11496 mertenslemi1 11498 zproddc 11542 dvdsfac 11820 |
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