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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 502 |
. . . 4
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2 | simpl2 953 |
. . . . 5
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3 | simpl3 954 |
. . . . 5
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4 | xrlenlt 7701 |
. . . . 5
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5 | 2, 3, 4 | syl2anc 406 |
. . . 4
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6 | 1, 5 | mpbid 146 |
. . 3
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7 | simprl 501 |
. . . 4
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8 | xrltso 9423 |
. . . . . 6
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9 | sowlin 4180 |
. . . . . 6
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10 | 8, 9 | mpan 418 |
. . . . 5
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11 | 10 | adantr 272 |
. . . 4
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12 | 7, 11 | mpd 13 |
. . 3
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13 | 6, 12 | ecased 1295 |
. 2
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14 | 13 | ex 114 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-po 4156 df-iso 4157 df-xp 4483 df-cnv 4485 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 |
This theorem is referenced by: xrltletrd 9435 xrre2 9445 xrre3 9446 ge0gtmnf 9447 iooss2 9541 iccssioo 9566 icossico 9567 icossioo 9588 ioossioo 9589 ioc0 9881 |
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