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Mirrors > Home > MPE Home > Th. List > icodisj | Structured version Visualization version 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 3796 | . . . 4 | |
2 | elico1 12218 | . . . . . . . . . 10 | |
3 | 2 | 3adant3 1081 | . . . . . . . . 9 |
4 | 3 | biimpa 501 | . . . . . . . 8 |
5 | 4 | simp3d 1075 | . . . . . . 7 |
6 | 5 | adantrr 753 | . . . . . 6 |
7 | elico1 12218 | . . . . . . . . . . 11 | |
8 | 7 | 3adant1 1079 | . . . . . . . . . 10 |
9 | 8 | biimpa 501 | . . . . . . . . 9 |
10 | 9 | simp2d 1074 | . . . . . . . 8 |
11 | simpl2 1065 | . . . . . . . . 9 | |
12 | 9 | simp1d 1073 | . . . . . . . . 9 |
13 | xrlenlt 10103 | . . . . . . . . 9 | |
14 | 11, 12, 13 | syl2anc 693 | . . . . . . . 8 |
15 | 10, 14 | mpbid 222 | . . . . . . 7 |
16 | 15 | adantrl 752 | . . . . . 6 |
17 | 6, 16 | pm2.65da 600 | . . . . 5 |
18 | 17 | pm2.21d 118 | . . . 4 |
19 | 1, 18 | syl5bi 232 | . . 3 |
20 | 19 | ssrdv 3609 | . 2 |
21 | ss0 3974 | . 2 | |
22 | 20, 21 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cin 3573 wss 3574 c0 3915 class class class wbr 4653 (class class class)co 6650 cxr 10073 clt 10074 cle 10075 cico 12177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-xr 10078 df-le 10080 df-ico 12181 |
This theorem is referenced by: icombl 23332 difico 29545 chtvalz 30707 |
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