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Mirrors > Home > ILE Home > Th. List > xrltletr | Unicode version |
Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
xrltletr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 521 | . . . 4 | |
2 | simpl2 985 | . . . . 5 | |
3 | simpl3 986 | . . . . 5 | |
4 | xrlenlt 7822 | . . . . 5 | |
5 | 2, 3, 4 | syl2anc 408 | . . . 4 |
6 | 1, 5 | mpbid 146 | . . 3 |
7 | simprl 520 | . . . 4 | |
8 | xrltso 9575 | . . . . . 6 | |
9 | sowlin 4237 | . . . . . 6 | |
10 | 8, 9 | mpan 420 | . . . . 5 |
11 | 10 | adantr 274 | . . . 4 |
12 | 7, 11 | mpd 13 | . . 3 |
13 | 6, 12 | ecased 1327 | . 2 |
14 | 13 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wcel 1480 class class class wbr 3924 wor 4212 cxr 7792 clt 7793 cle 7794 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-po 4213 df-iso 4214 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: xrltletrd 9587 xrre2 9597 xrre3 9598 ge0gtmnf 9599 iooss2 9693 iccssioo 9718 icossico 9719 icossioo 9740 ioossioo 9741 ioc0 10033 |
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