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Mirrors > Home > ILE Home > Th. List > xrltnr | Unicode version |
Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
xrltnr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9593 |
. 2
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2 | ltnr 7865 |
. . 3
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3 | pnfnre 7831 |
. . . . . . . . . 10
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4 | 3 | neli 2406 |
. . . . . . . . 9
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5 | 4 | intnan 915 |
. . . . . . . 8
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6 | 5 | intnanr 916 |
. . . . . . 7
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7 | pnfnemnf 7844 |
. . . . . . . . 9
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8 | 7 | neii 2311 |
. . . . . . . 8
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9 | 8 | intnanr 916 |
. . . . . . 7
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10 | 6, 9 | pm3.2ni 803 |
. . . . . 6
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11 | 4 | intnanr 916 |
. . . . . . 7
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12 | 4 | intnan 915 |
. . . . . . 7
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13 | 11, 12 | pm3.2ni 803 |
. . . . . 6
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14 | 10, 13 | pm3.2ni 803 |
. . . . 5
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15 | pnfxr 7842 |
. . . . . 6
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16 | ltxr 9592 |
. . . . . 6
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17 | 15, 15, 16 | mp2an 423 |
. . . . 5
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18 | 14, 17 | mtbir 661 |
. . . 4
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19 | breq12 3942 |
. . . . 5
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20 | 19 | anidms 395 |
. . . 4
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21 | 18, 20 | mtbiri 665 |
. . 3
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22 | mnfnre 7832 |
. . . . . . . . . 10
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23 | 22 | neli 2406 |
. . . . . . . . 9
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24 | 23 | intnan 915 |
. . . . . . . 8
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25 | 24 | intnanr 916 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 7 | nesymi 2355 |
. . . . . . . 8
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27 | 26 | intnan 915 |
. . . . . . 7
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28 | 25, 27 | pm3.2ni 803 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 23 | intnanr 916 |
. . . . . . 7
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30 | 23 | intnan 915 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 29, 30 | pm3.2ni 803 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 28, 31 | pm3.2ni 803 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | mnfxr 7846 |
. . . . . 6
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34 | ltxr 9592 |
. . . . . 6
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35 | 33, 33, 34 | mp2an 423 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 32, 35 | mtbir 661 |
. . . 4
![]() ![]() ![]() ![]() ![]() |
37 | breq12 3942 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 37 | anidms 395 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 36, 38 | mtbiri 665 |
. . 3
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40 | 2, 21, 39 | 3jaoi 1282 |
. 2
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41 | 1, 40 | sylbi 120 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 |
This theorem is referenced by: xrltnsym 9609 xrltso 9612 xrlttri3 9613 xrleid 9616 xrltne 9626 nltpnft 9627 ngtmnft 9630 xrrebnd 9632 xposdif 9695 lbioog 9726 ubioog 9727 xrmaxleim 11045 xrmaxiflemlub 11049 |
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