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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 8696 | . 2 | |
2 | elxr 8696 | . 2 | |
3 | ltnsym 7104 | . . . 4 | |
4 | rexr 7071 | . . . . . . . 8 | |
5 | pnfnlt 8708 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | 6 | adantr 261 | . . . . . 6 |
8 | breq1 3767 | . . . . . . 7 | |
9 | 8 | adantl 262 | . . . . . 6 |
10 | 7, 9 | mtbird 598 | . . . . 5 |
11 | 10 | a1d 22 | . . . 4 |
12 | nltmnf 8709 | . . . . . . . 8 | |
13 | 4, 12 | syl 14 | . . . . . . 7 |
14 | 13 | adantr 261 | . . . . . 6 |
15 | breq2 3768 | . . . . . . 7 | |
16 | 15 | adantl 262 | . . . . . 6 |
17 | 14, 16 | mtbird 598 | . . . . 5 |
18 | 17 | pm2.21d 549 | . . . 4 |
19 | 3, 11, 18 | 3jaodan 1201 | . . 3 |
20 | pnfnlt 8708 | . . . . . . 7 | |
21 | 20 | adantl 262 | . . . . . 6 |
22 | breq1 3767 | . . . . . . 7 | |
23 | 22 | adantr 261 | . . . . . 6 |
24 | 21, 23 | mtbird 598 | . . . . 5 |
25 | 24 | pm2.21d 549 | . . . 4 |
26 | 2, 25 | sylan2br 272 | . . 3 |
27 | rexr 7071 | . . . . . . . 8 | |
28 | nltmnf 8709 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | 29 | adantl 262 | . . . . . 6 |
31 | breq2 3768 | . . . . . . 7 | |
32 | 31 | adantr 261 | . . . . . 6 |
33 | 30, 32 | mtbird 598 | . . . . 5 |
34 | 33 | a1d 22 | . . . 4 |
35 | mnfxr 8694 | . . . . . . . 8 | |
36 | pnfnlt 8708 | . . . . . . . 8 | |
37 | 35, 36 | ax-mp 7 | . . . . . . 7 |
38 | breq12 3769 | . . . . . . 7 | |
39 | 37, 38 | mtbiri 600 | . . . . . 6 |
40 | 39 | ancoms 255 | . . . . 5 |
41 | 40 | a1d 22 | . . . 4 |
42 | xrltnr 8701 | . . . . . . 7 | |
43 | 35, 42 | ax-mp 7 | . . . . . 6 |
44 | breq12 3769 | . . . . . 6 | |
45 | 43, 44 | mtbiri 600 | . . . . 5 |
46 | 45 | pm2.21d 549 | . . . 4 |
47 | 34, 41, 46 | 3jaodan 1201 | . . 3 |
48 | 19, 26, 47 | 3jaoian 1200 | . 2 |
49 | 1, 2, 48 | syl2anb 275 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 w3o 884 wceq 1243 wcel 1393 class class class wbr 3764 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-lttrn 6998 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: xrltnsym2 8715 xrltle 8719 |
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