Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 521 | . . . . 5 | |
2 | simpl1 985 | . . . . . 6 | |
3 | simpl2 986 | . . . . . 6 | |
4 | xrlenlt 7942 | . . . . . 6 | |
5 | 2, 3, 4 | syl2anc 409 | . . . . 5 |
6 | 1, 5 | mpbid 146 | . . . 4 |
7 | 6 | pm2.21d 609 | . . 3 |
8 | idd 21 | . . 3 | |
9 | simprr 522 | . . . 4 | |
10 | simpl3 987 | . . . . 5 | |
11 | xrltso 9703 | . . . . . 6 | |
12 | sowlin 4280 | . . . . . 6 | |
13 | 11, 12 | mpan 421 | . . . . 5 |
14 | 3, 10, 2, 13 | syl3anc 1220 | . . . 4 |
15 | 9, 14 | mpd 13 | . . 3 |
16 | 7, 8, 15 | mpjaod 708 | . 2 |
17 | 16 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wcel 2128 class class class wbr 3965 wor 4255 cxr 7911 clt 7912 cle 7913 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 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-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 3773 df-br 3966 df-opab 4026 df-po 4256 df-iso 4257 df-xp 4592 df-cnv 4594 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 |
This theorem is referenced by: xrlelttrd 9714 xrre 9724 xrre2 9725 iooss1 9820 iccssioo 9846 iccssico 9849 iocssioo 9867 ioossioo 9869 ico0 10161 bldisj 12812 xblm 12828 blsscls2 12904 metcnpi3 12928 |
Copyright terms: Public domain | W3C validator |