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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 9445 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | eluzelz 9449 | . . 3 | |
4 | 3 | adantr 274 | . 2 |
5 | eluzle 9452 | . . . 4 | |
6 | 5 | adantl 275 | . . 3 |
7 | eluzle 9452 | . . . 4 | |
8 | 7 | adantr 274 | . . 3 |
9 | eluzelz 9449 | . . . . 5 | |
10 | 9 | adantl 275 | . . . 4 |
11 | zletr 9217 | . . . 4 | |
12 | 2, 10, 4, 11 | syl3anc 1220 | . . 3 |
13 | 6, 8, 12 | mp2and 430 | . 2 |
14 | eluz2 9446 | . 2 | |
15 | 2, 4, 13, 14 | syl3anbrc 1166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 class class class wbr 3966 cfv 5171 cle 7914 cz 9168 cuz 9440 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltwlin 7846 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-neg 8050 df-z 9169 df-uz 9441 |
This theorem is referenced by: uztrn2 9457 fzsplit2 9953 fzass4 9965 fzss1 9966 fzss2 9967 uzsplit 9995 seq3fveq2 10372 ser3mono 10381 seq3split 10382 seq3f1olemqsumkj 10401 seq3f1olemqsumk 10402 seq3id 10411 seq3id2 10412 seq3z 10414 seq3coll 10717 cvgratgt0 11434 mertenslemi1 11436 zproddc 11480 dvdsfac 11756 |
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