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Mirrors > Home > ILE Home > Th. List > xrltnsym | Unicode version |
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
xrltnsym |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9593 |
. 2
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2 | elxr 9593 |
. 2
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3 | ltnsym 7874 |
. . . 4
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4 | rexr 7835 |
. . . . . . . 8
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5 | pnfnlt 9603 |
. . . . . . . 8
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6 | 4, 5 | syl 14 |
. . . . . . 7
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7 | 6 | adantr 274 |
. . . . . 6
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8 | breq1 3940 |
. . . . . . 7
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9 | 8 | adantl 275 |
. . . . . 6
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10 | 7, 9 | mtbird 663 |
. . . . 5
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11 | 10 | a1d 22 |
. . . 4
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12 | nltmnf 9604 |
. . . . . . . 8
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13 | 4, 12 | syl 14 |
. . . . . . 7
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14 | 13 | adantr 274 |
. . . . . 6
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15 | breq2 3941 |
. . . . . . 7
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16 | 15 | adantl 275 |
. . . . . 6
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17 | 14, 16 | mtbird 663 |
. . . . 5
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18 | 17 | pm2.21d 609 |
. . . 4
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19 | 3, 11, 18 | 3jaodan 1285 |
. . 3
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20 | pnfnlt 9603 |
. . . . . . 7
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21 | 20 | adantl 275 |
. . . . . 6
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22 | breq1 3940 |
. . . . . . 7
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23 | 22 | adantr 274 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | mtbird 663 |
. . . . 5
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25 | 24 | pm2.21d 609 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 2, 25 | sylan2br 286 |
. . 3
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27 | rexr 7835 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | nltmnf 9604 |
. . . . . . . 8
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29 | 27, 28 | syl 14 |
. . . . . . 7
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30 | 29 | adantl 275 |
. . . . . 6
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31 | breq2 3941 |
. . . . . . 7
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32 | 31 | adantr 274 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 30, 32 | mtbird 663 |
. . . . 5
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34 | 33 | a1d 22 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | mnfxr 7846 |
. . . . . . . 8
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36 | pnfnlt 9603 |
. . . . . . . 8
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37 | 35, 36 | ax-mp 5 |
. . . . . . 7
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38 | breq12 3942 |
. . . . . . 7
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39 | 37, 38 | mtbiri 665 |
. . . . . 6
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40 | 39 | ancoms 266 |
. . . . 5
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41 | 40 | a1d 22 |
. . . 4
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42 | xrltnr 9596 |
. . . . . . 7
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43 | 35, 42 | ax-mp 5 |
. . . . . 6
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44 | breq12 3942 |
. . . . . 6
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45 | 43, 44 | mtbiri 665 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 45 | pm2.21d 609 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 34, 41, 46 | 3jaodan 1285 |
. . 3
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48 | 19, 26, 47 | 3jaoian 1284 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 1, 2, 48 | syl2anb 289 |
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 ax-pre-lttrn 7758 |
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: xrltnsym2 9610 xrltle 9614 |
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