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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 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elxr 9898 |
. 2
| |
| 2 | elxr 9898 |
. 2
| |
| 3 | ltnsym 8158 |
. . . 4
| |
| 4 | rexr 8118 |
. . . . . . . 8
| |
| 5 | pnfnlt 9909 |
. . . . . . . 8
| |
| 6 | 4, 5 | syl 14 |
. . . . . . 7
|
| 7 | 6 | adantr 276 |
. . . . . 6
|
| 8 | breq1 4047 |
. . . . . . 7
| |
| 9 | 8 | adantl 277 |
. . . . . 6
|
| 10 | 7, 9 | mtbird 675 |
. . . . 5
|
| 11 | 10 | a1d 22 |
. . . 4
|
| 12 | nltmnf 9910 |
. . . . . . . 8
| |
| 13 | 4, 12 | syl 14 |
. . . . . . 7
|
| 14 | 13 | adantr 276 |
. . . . . 6
|
| 15 | breq2 4048 |
. . . . . . 7
| |
| 16 | 15 | adantl 277 |
. . . . . 6
|
| 17 | 14, 16 | mtbird 675 |
. . . . 5
|
| 18 | 17 | pm2.21d 620 |
. . . 4
|
| 19 | 3, 11, 18 | 3jaodan 1319 |
. . 3
|
| 20 | pnfnlt 9909 |
. . . . . . 7
| |
| 21 | 20 | adantl 277 |
. . . . . 6
|
| 22 | breq1 4047 |
. . . . . . 7
| |
| 23 | 22 | adantr 276 |
. . . . . 6
|
| 24 | 21, 23 | mtbird 675 |
. . . . 5
|
| 25 | 24 | pm2.21d 620 |
. . . 4
|
| 26 | 2, 25 | sylan2br 288 |
. . 3
|
| 27 | rexr 8118 |
. . . . . . . 8
| |
| 28 | nltmnf 9910 |
. . . . . . . 8
| |
| 29 | 27, 28 | syl 14 |
. . . . . . 7
|
| 30 | 29 | adantl 277 |
. . . . . 6
|
| 31 | breq2 4048 |
. . . . . . 7
| |
| 32 | 31 | adantr 276 |
. . . . . 6
|
| 33 | 30, 32 | mtbird 675 |
. . . . 5
|
| 34 | 33 | a1d 22 |
. . . 4
|
| 35 | mnfxr 8129 |
. . . . . . . 8
| |
| 36 | pnfnlt 9909 |
. . . . . . . 8
| |
| 37 | 35, 36 | ax-mp 5 |
. . . . . . 7
|
| 38 | breq12 4049 |
. . . . . . 7
| |
| 39 | 37, 38 | mtbiri 677 |
. . . . . 6
|
| 40 | 39 | ancoms 268 |
. . . . 5
|
| 41 | 40 | a1d 22 |
. . . 4
|
| 42 | xrltnr 9901 |
. . . . . . 7
| |
| 43 | 35, 42 | ax-mp 5 |
. . . . . 6
|
| 44 | breq12 4049 |
. . . . . 6
| |
| 45 | 43, 44 | mtbiri 677 |
. . . . 5
|
| 46 | 45 | pm2.21d 620 |
. . . 4
|
| 47 | 34, 41, 46 | 3jaodan 1319 |
. . 3
|
| 48 | 19, 26, 47 | 3jaoian 1318 |
. 2
|
| 49 | 1, 2, 48 | syl2anb 291 |
1
|
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-pre-ltirr 8037 ax-pre-lttrn 8039 |
| This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-xp 4681 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 |
| This theorem is referenced by: xrltnsym2 9916 xrltle 9920 |
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