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Mirrors > Home > ILE Home > Th. List > apsym | Unicode version |
Description: Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
Ref | Expression |
---|---|
apsym | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7023 | . . 3 | |
2 | 1 | adantl 262 | . 2 |
3 | cnre 7023 | . . . . . 6 | |
4 | 3 | ad3antrrr 461 | . . . . 5 |
5 | simplrl 487 | . . . . . . . . . . . 12 | |
6 | simplrl 487 | . . . . . . . . . . . . 13 | |
7 | 6 | ad2antrr 457 | . . . . . . . . . . . 12 |
8 | reaplt 7579 | . . . . . . . . . . . 12 # | |
9 | 5, 7, 8 | syl2anc 391 | . . . . . . . . . . 11 # |
10 | reaplt 7579 | . . . . . . . . . . . . 13 # | |
11 | 7, 5, 10 | syl2anc 391 | . . . . . . . . . . . 12 # |
12 | orcom 647 | . . . . . . . . . . . 12 | |
13 | 11, 12 | syl6bbr 187 | . . . . . . . . . . 11 # |
14 | 9, 13 | bitr4d 180 | . . . . . . . . . 10 # # |
15 | simplrr 488 | . . . . . . . . . . . 12 | |
16 | simplrr 488 | . . . . . . . . . . . . 13 | |
17 | 16 | ad2antrr 457 | . . . . . . . . . . . 12 |
18 | reaplt 7579 | . . . . . . . . . . . 12 # | |
19 | 15, 17, 18 | syl2anc 391 | . . . . . . . . . . 11 # |
20 | reaplt 7579 | . . . . . . . . . . . . 13 # | |
21 | 17, 15, 20 | syl2anc 391 | . . . . . . . . . . . 12 # |
22 | orcom 647 | . . . . . . . . . . . 12 | |
23 | 21, 22 | syl6bbr 187 | . . . . . . . . . . 11 # |
24 | 19, 23 | bitr4d 180 | . . . . . . . . . 10 # # |
25 | 14, 24 | orbi12d 707 | . . . . . . . . 9 # # # # |
26 | apreim 7594 | . . . . . . . . . 10 # # # | |
27 | 5, 15, 7, 17, 26 | syl22anc 1136 | . . . . . . . . 9 # # # |
28 | apreim 7594 | . . . . . . . . . 10 # # # | |
29 | 7, 17, 5, 15, 28 | syl22anc 1136 | . . . . . . . . 9 # # # |
30 | 25, 27, 29 | 3bitr4d 209 | . . . . . . . 8 # # |
31 | simpr 103 | . . . . . . . . 9 | |
32 | simpllr 486 | . . . . . . . . 9 | |
33 | 31, 32 | breq12d 3777 | . . . . . . . 8 # # |
34 | 32, 31 | breq12d 3777 | . . . . . . . 8 # # |
35 | 30, 33, 34 | 3bitr4d 209 | . . . . . . 7 # # |
36 | 35 | ex 108 | . . . . . 6 # # |
37 | 36 | rexlimdvva 2440 | . . . . 5 # # |
38 | 4, 37 | mpd 13 | . . . 4 # # |
39 | 38 | ex 108 | . . 3 # # |
40 | 39 | rexlimdvva 2440 | . 2 # # |
41 | 2, 40 | mpd 13 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wrex 2307 class class class wbr 3764 (class class class)co 5512 cc 6887 cr 6888 ci 6891 caddc 6892 cmul 6894 clt 7060 # cap 7572 |
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-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 |
This theorem depends on definitions: df-bi 110 df-dc 743 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-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 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-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-ltxr 7065 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 |
This theorem is referenced by: addext 7601 mulext 7605 ltapii 7624 ltapd 7627 recgt0 7816 prodgt0 7818 |
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