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Mirrors > Home > ILE Home > Th. List > ltntri | Unicode version |
Description: Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, . Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.) |
Ref | Expression |
---|---|
ltntri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 524 | . . . 4 | |
2 | simplr 525 | . . . 4 | |
3 | simpr3 1000 | . . . 4 | |
4 | 1, 2, 3 | nltled 8033 | . . 3 |
5 | simpr1 998 | . . . 4 | |
6 | 2, 1, 5 | nltled 8033 | . . 3 |
7 | 1, 2 | letri3d 8028 | . . 3 |
8 | 4, 6, 7 | mpbir2and 939 | . 2 |
9 | simpr2 999 | . 2 | |
10 | 8, 9 | pm2.65da 656 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cr 7766 clt 7947 cle 7948 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-pre-ltirr 7879 ax-pre-apti 7882 |
This theorem depends on definitions: df-bi 116 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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 |
This theorem is referenced by: (None) |
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