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Mirrors > Home > ILE Home > Th. List > xrlelttr | Unicode version |
Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
xrlelttr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 526 | . . . . 5 | |
2 | simpl1 995 | . . . . . 6 | |
3 | simpl2 996 | . . . . . 6 | |
4 | xrlenlt 7984 | . . . . . 6 | |
5 | 2, 3, 4 | syl2anc 409 | . . . . 5 |
6 | 1, 5 | mpbid 146 | . . . 4 |
7 | 6 | pm2.21d 614 | . . 3 |
8 | idd 21 | . . 3 | |
9 | simprr 527 | . . . 4 | |
10 | simpl3 997 | . . . . 5 | |
11 | xrltso 9753 | . . . . . 6 | |
12 | sowlin 4305 | . . . . . 6 | |
13 | 11, 12 | mpan 422 | . . . . 5 |
14 | 3, 10, 2, 13 | syl3anc 1233 | . . . 4 |
15 | 9, 14 | mpd 13 | . . 3 |
16 | 7, 8, 15 | mpjaod 713 | . 2 |
17 | 16 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wcel 2141 class class class wbr 3989 wor 4280 cxr 7953 clt 7954 cle 7955 |
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-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 |
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-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-po 4281 df-iso 4282 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 |
This theorem is referenced by: xrlelttrd 9767 xrre 9777 xrre2 9778 iooss1 9873 iccssioo 9899 iccssico 9902 iocssioo 9920 ioossioo 9922 ico0 10218 bldisj 13195 xblm 13211 blsscls2 13287 metcnpi3 13311 |
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