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Mirrors > Home > ILE Home > Th. List > ixxdisj | Unicode version |
Description: Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
ixxssixx.1 | |
ixxun.2 | |
ixxun.3 |
Ref | Expression |
---|---|
ixxdisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3259 | . . . 4 | |
2 | ixxssixx.1 | . . . . . . . . . . 11 | |
3 | 2 | elixx1 9680 | . . . . . . . . . 10 |
4 | 3 | 3adant3 1001 | . . . . . . . . 9 |
5 | 4 | biimpa 294 | . . . . . . . 8 |
6 | 5 | simp3d 995 | . . . . . . 7 |
7 | 6 | adantrr 470 | . . . . . 6 |
8 | ixxun.2 | . . . . . . . . . . . 12 | |
9 | 8 | elixx1 9680 | . . . . . . . . . . 11 |
10 | 9 | 3adant1 999 | . . . . . . . . . 10 |
11 | 10 | biimpa 294 | . . . . . . . . 9 |
12 | 11 | simp2d 994 | . . . . . . . 8 |
13 | simpl2 985 | . . . . . . . . 9 | |
14 | 11 | simp1d 993 | . . . . . . . . 9 |
15 | ixxun.3 | . . . . . . . . 9 | |
16 | 13, 14, 15 | syl2anc 408 | . . . . . . . 8 |
17 | 12, 16 | mpbid 146 | . . . . . . 7 |
18 | 17 | adantrl 469 | . . . . . 6 |
19 | 7, 18 | pm2.65da 650 | . . . . 5 |
20 | 19 | pm2.21d 608 | . . . 4 |
21 | 1, 20 | syl5bi 151 | . . 3 |
22 | 21 | ssrdv 3103 | . 2 |
23 | ss0 3403 | . 2 | |
24 | 22, 23 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 crab 2420 cin 3070 wss 3071 c0 3363 class class class wbr 3929 (class class class)co 5774 cmpo 5776 cxr 7799 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 |
This theorem is referenced by: ioodisj 9776 |
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