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Mirrors > Home > ILE Home > Th. List > xrlelttrd | Unicode version |
Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xrlttrd.1 |
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xrlttrd.2 |
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xrlttrd.3 |
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xrlelttrd.4 |
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xrlelttrd.5 |
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Ref | Expression |
---|---|
xrlelttrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlelttrd.4 |
. 2
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2 | xrlelttrd.5 |
. 2
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3 | xrlttrd.1 |
. . 3
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4 | xrlttrd.2 |
. . 3
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5 | xrlttrd.3 |
. . 3
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6 | xrlelttr 9842 |
. . 3
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7 | 3, 4, 5, 6 | syl3anc 1249 |
. 2
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8 | 1, 2, 7 | mp2and 433 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-po 4317 df-iso 4318 df-xp 4653 df-cnv 4655 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 |
This theorem is referenced by: xlt2add 9916 elioc2 9972 elicc2 9974 xrmaxltsup 11307 blgt0 14387 xblss2ps 14389 xblss2 14390 tgioo 14531 |
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