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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 9684 | . 2 | |
2 | elxr 9684 | . 2 | |
3 | ltnsym 7964 | . . . 4 | |
4 | rexr 7924 | . . . . . . . 8 | |
5 | pnfnlt 9695 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | 6 | adantr 274 | . . . . . 6 |
8 | breq1 3969 | . . . . . . 7 | |
9 | 8 | adantl 275 | . . . . . 6 |
10 | 7, 9 | mtbird 663 | . . . . 5 |
11 | 10 | a1d 22 | . . . 4 |
12 | nltmnf 9696 | . . . . . . . 8 | |
13 | 4, 12 | syl 14 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | breq2 3970 | . . . . . . 7 | |
16 | 15 | adantl 275 | . . . . . 6 |
17 | 14, 16 | mtbird 663 | . . . . 5 |
18 | 17 | pm2.21d 609 | . . . 4 |
19 | 3, 11, 18 | 3jaodan 1288 | . . 3 |
20 | pnfnlt 9695 | . . . . . . 7 | |
21 | 20 | adantl 275 | . . . . . 6 |
22 | breq1 3969 | . . . . . . 7 | |
23 | 22 | adantr 274 | . . . . . 6 |
24 | 21, 23 | mtbird 663 | . . . . 5 |
25 | 24 | pm2.21d 609 | . . . 4 |
26 | 2, 25 | sylan2br 286 | . . 3 |
27 | rexr 7924 | . . . . . . . 8 | |
28 | nltmnf 9696 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | 29 | adantl 275 | . . . . . 6 |
31 | breq2 3970 | . . . . . . 7 | |
32 | 31 | adantr 274 | . . . . . 6 |
33 | 30, 32 | mtbird 663 | . . . . 5 |
34 | 33 | a1d 22 | . . . 4 |
35 | mnfxr 7935 | . . . . . . . 8 | |
36 | pnfnlt 9695 | . . . . . . . 8 | |
37 | 35, 36 | ax-mp 5 | . . . . . . 7 |
38 | breq12 3971 | . . . . . . 7 | |
39 | 37, 38 | mtbiri 665 | . . . . . 6 |
40 | 39 | ancoms 266 | . . . . 5 |
41 | 40 | a1d 22 | . . . 4 |
42 | xrltnr 9687 | . . . . . . 7 | |
43 | 35, 42 | ax-mp 5 | . . . . . 6 |
44 | breq12 3971 | . . . . . 6 | |
45 | 43, 44 | mtbiri 665 | . . . . 5 |
46 | 45 | pm2.21d 609 | . . . 4 |
47 | 34, 41, 46 | 3jaodan 1288 | . . 3 |
48 | 19, 26, 47 | 3jaoian 1287 | . 2 |
49 | 1, 2, 48 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 962 wceq 1335 wcel 2128 class class class wbr 3966 cr 7732 cpnf 7910 cmnf 7911 cxr 7912 clt 7913 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltirr 7845 ax-pre-lttrn 7847 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 |
This theorem is referenced by: xrltnsym2 9702 xrltle 9706 |
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