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Mirrors > Home > ILE Home > Th. List > icodisj | Unicode version |
Description: End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
icodisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3126 | . . . 4 | |
2 | elico1 8792 | . . . . . . . . . 10 | |
3 | 2 | 3adant3 924 | . . . . . . . . 9 |
4 | 3 | biimpa 280 | . . . . . . . 8 |
5 | 4 | simp3d 918 | . . . . . . 7 |
6 | 5 | adantrr 448 | . . . . . 6 |
7 | elico1 8792 | . . . . . . . . . . 11 | |
8 | 7 | 3adant1 922 | . . . . . . . . . 10 |
9 | 8 | biimpa 280 | . . . . . . . . 9 |
10 | 9 | simp2d 917 | . . . . . . . 8 |
11 | simpl2 908 | . . . . . . . . 9 | |
12 | 9 | simp1d 916 | . . . . . . . . 9 |
13 | xrlenlt 7084 | . . . . . . . . 9 | |
14 | 11, 12, 13 | syl2anc 391 | . . . . . . . 8 |
15 | 10, 14 | mpbid 135 | . . . . . . 7 |
16 | 15 | adantrl 447 | . . . . . 6 |
17 | 6, 16 | pm2.65da 587 | . . . . 5 |
18 | 17 | pm2.21d 549 | . . . 4 |
19 | 1, 18 | syl5bi 141 | . . 3 |
20 | 19 | ssrdv 2951 | . 2 |
21 | ss0 3257 | . 2 | |
22 | 20, 21 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cin 2916 wss 2917 c0 3224 class class class wbr 3764 (class class class)co 5512 cxr 7059 clt 7060 cle 7061 cico 8759 |
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 |
This theorem depends on definitions: df-bi 110 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-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-le 7066 df-ico 8763 |
This theorem is referenced by: (None) |
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