Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8850 | . 2 | |
2 | elxr 8850 | . 2 | |
3 | ltnsym 7197 | . . . 4 | |
4 | rexr 7164 | . . . . . . . 8 | |
5 | pnfnlt 8862 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | 6 | adantr 270 | . . . . . 6 |
8 | breq1 3788 | . . . . . . 7 | |
9 | 8 | adantl 271 | . . . . . 6 |
10 | 7, 9 | mtbird 630 | . . . . 5 |
11 | 10 | a1d 22 | . . . 4 |
12 | nltmnf 8863 | . . . . . . . 8 | |
13 | 4, 12 | syl 14 | . . . . . . 7 |
14 | 13 | adantr 270 | . . . . . 6 |
15 | breq2 3789 | . . . . . . 7 | |
16 | 15 | adantl 271 | . . . . . 6 |
17 | 14, 16 | mtbird 630 | . . . . 5 |
18 | 17 | pm2.21d 581 | . . . 4 |
19 | 3, 11, 18 | 3jaodan 1237 | . . 3 |
20 | pnfnlt 8862 | . . . . . . 7 | |
21 | 20 | adantl 271 | . . . . . 6 |
22 | breq1 3788 | . . . . . . 7 | |
23 | 22 | adantr 270 | . . . . . 6 |
24 | 21, 23 | mtbird 630 | . . . . 5 |
25 | 24 | pm2.21d 581 | . . . 4 |
26 | 2, 25 | sylan2br 282 | . . 3 |
27 | rexr 7164 | . . . . . . . 8 | |
28 | nltmnf 8863 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | 29 | adantl 271 | . . . . . 6 |
31 | breq2 3789 | . . . . . . 7 | |
32 | 31 | adantr 270 | . . . . . 6 |
33 | 30, 32 | mtbird 630 | . . . . 5 |
34 | 33 | a1d 22 | . . . 4 |
35 | mnfxr 8848 | . . . . . . . 8 | |
36 | pnfnlt 8862 | . . . . . . . 8 | |
37 | 35, 36 | ax-mp 7 | . . . . . . 7 |
38 | breq12 3790 | . . . . . . 7 | |
39 | 37, 38 | mtbiri 632 | . . . . . 6 |
40 | 39 | ancoms 264 | . . . . 5 |
41 | 40 | a1d 22 | . . . 4 |
42 | xrltnr 8855 | . . . . . . 7 | |
43 | 35, 42 | ax-mp 7 | . . . . . 6 |
44 | breq12 3790 | . . . . . 6 | |
45 | 43, 44 | mtbiri 632 | . . . . 5 |
46 | 45 | pm2.21d 581 | . . . 4 |
47 | 34, 41, 46 | 3jaodan 1237 | . . 3 |
48 | 19, 26, 47 | 3jaoian 1236 | . 2 |
49 | 1, 2, 48 | syl2anb 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3o 918 wceq 1284 wcel 1433 class class class wbr 3785 cr 6980 cpnf 7150 cmnf 7151 cxr 7152 clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 |
This theorem is referenced by: xrltnsym2 8869 xrltle 8873 |
Copyright terms: Public domain | W3C validator |