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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnoutside | Structured version Visualization version Unicode version |
Description: A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
btwnoutside | OutsideOf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1039 | . . . . . 6 | |
2 | simpr11 1145 | . . . . . . 7 | |
3 | simpr12 1146 | . . . . . . 7 | |
4 | simpr13 1147 | . . . . . . . 8 | |
5 | simp1 1061 | . . . . . . . . 9 | |
6 | simp3r 1090 | . . . . . . . . 9 | |
7 | simp2l 1087 | . . . . . . . . 9 | |
8 | simp3l 1089 | . . . . . . . . 9 | |
9 | simpr2 1068 | . . . . . . . . 9 | |
10 | 5, 6, 7, 8, 9 | btwncomand 32122 | . . . . . . . 8 |
11 | simp2r 1088 | . . . . . . . . 9 | |
12 | simpr3 1069 | . . . . . . . . 9 | |
13 | 5, 6, 11, 8, 12 | btwncomand 32122 | . . . . . . . 8 |
14 | btwnconn2 32209 | . . . . . . . . . 10 | |
15 | 14 | 3com23 1271 | . . . . . . . . 9 |
16 | 15 | adantr 481 | . . . . . . . 8 |
17 | 4, 10, 13, 16 | mp3and 1427 | . . . . . . 7 |
18 | 2, 3, 17 | 3jca 1242 | . . . . . 6 |
19 | 1, 18 | sylan2br 493 | . . . . 5 |
20 | 19 | expr 643 | . . . 4 |
21 | simp3 1063 | . . . . 5 | |
22 | df-3an 1039 | . . . . . . . 8 | |
23 | simpr11 1145 | . . . . . . . . 9 | |
24 | simpr3 1069 | . . . . . . . . . 10 | |
25 | 5, 7, 6, 11, 24 | btwncomand 32122 | . . . . . . . . 9 |
26 | simpr2 1068 | . . . . . . . . 9 | |
27 | btwnouttr2 32129 | . . . . . . . . . . 11 | |
28 | 5, 11, 7, 6, 8, 27 | syl122anc 1335 | . . . . . . . . . 10 |
29 | 28 | adantr 481 | . . . . . . . . 9 |
30 | 23, 25, 26, 29 | mp3and 1427 | . . . . . . . 8 |
31 | 22, 30 | sylan2br 493 | . . . . . . 7 |
32 | 31 | expr 643 | . . . . . 6 |
33 | df-3an 1039 | . . . . . . . 8 | |
34 | simpr3 1069 | . . . . . . . . . 10 | |
35 | 5, 11, 6, 7, 34 | btwncomand 32122 | . . . . . . . . 9 |
36 | simpr2 1068 | . . . . . . . . 9 | |
37 | 5, 7, 11, 6, 8, 35, 36 | btwnexch3and 32128 | . . . . . . . 8 |
38 | 33, 37 | sylan2br 493 | . . . . . . 7 |
39 | 38 | expr 643 | . . . . . 6 |
40 | 32, 39 | jaod 395 | . . . . 5 |
41 | 21, 40 | syl5 34 | . . . 4 |
42 | 20, 41 | impbid 202 | . . 3 |
43 | broutsideof2 32229 | . . . . 5 OutsideOf | |
44 | 5, 6, 7, 11, 43 | syl13anc 1328 | . . . 4 OutsideOf |
45 | 44 | adantr 481 | . . 3 OutsideOf |
46 | 42, 45 | bitr4d 271 | . 2 OutsideOf |
47 | 46 | ex 450 | 1 OutsideOf |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 w3a 1037 wcel 1990 wne 2794 cop 4183 class class class wbr 4653 cfv 5888 cn 11020 cee 25768 cbtwn 25769 OutsideOfcoutsideof 32226 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-ee 25771 df-btwn 25772 df-cgr 25773 df-ofs 32090 df-colinear 32146 df-ifs 32147 df-cgr3 32148 df-fs 32149 df-outsideof 32227 |
This theorem is referenced by: (None) |
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