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Mirrors > Home > ILE Home > Th. List > xrre2 | Unicode version |
Description: An extended real between two others is real. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
xrre2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfle 8713 | . . . . . . 7 | |
2 | 1 | adantr 261 | . . . . . 6 |
3 | mnfxr 8694 | . . . . . . 7 | |
4 | xrlelttr 8722 | . . . . . . 7 | |
5 | 3, 4 | mp3an1 1219 | . . . . . 6 |
6 | 2, 5 | mpand 405 | . . . . 5 |
7 | 6 | 3adant3 924 | . . . 4 |
8 | pnfge 8710 | . . . . . . 7 | |
9 | 8 | adantl 262 | . . . . . 6 |
10 | pnfxr 8692 | . . . . . . 7 | |
11 | xrltletr 8723 | . . . . . . 7 | |
12 | 10, 11 | mp3an3 1221 | . . . . . 6 |
13 | 9, 12 | mpan2d 404 | . . . . 5 |
14 | 13 | 3adant1 922 | . . . 4 |
15 | 7, 14 | anim12d 318 | . . 3 |
16 | xrrebnd 8732 | . . . 4 | |
17 | 16 | 3ad2ant2 926 | . . 3 |
18 | 15, 17 | sylibrd 158 | . 2 |
19 | 18 | imp 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 class class class wbr 3764 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 cle 7061 |
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-ltwlin 6997 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-po 4033 df-iso 4034 df-xp 4351 df-cnv 4353 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: elioore 8781 |
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