apcotr - Intuitionistic Logic Explorer

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Theorem apcotr 7598

Description: Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)

**Assertion**
Ref	Expression
apcotr	$/\$ $/\$ # # $\/$ #

Proof of Theorem apcotr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 7023	. . 3
2	1	3ad2ant3 927	. 2 $/\$ $/\$
3		cnre 7023	. . . . . . 7
4	3	3ad2ant2 926	. . . . . 6 $/\$ $/\$
5	4	ad2antrr 457	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
6		cnre 7023	. . . . . . . . . . 11
7	6	3ad2ant1 925	. . . . . . . . . 10 $/\$ $/\$
8	7	adantr 261	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
9	8	ad3antrrr 461	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpr 103	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11		simpllr 486	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	breq12d 3777	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
13		simplrl 487	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplrr 488	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simprl 483	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	ad3antrrr 461	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		simprr 484	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ad3antrrr 461	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		apreim 7594	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ # # $\/$ #
20	13, 14, 16, 18, 19	syl22anc 1136	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
21	12, 20	bitrd 177	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
22		simprl 483	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
23	22	ad2antrr 457	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	23	ad3antrrr 461	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		reapcotr 7589	. . . . . . . . . . . . . . 15 $/\$ $/\$ # # $\/$ #
26	13, 16, 24, 25	syl3anc 1135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
27		simprr 484	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
28	27	ad2antrr 457	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ad3antrrr 461	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		reapcotr 7589	. . . . . . . . . . . . . . 15 $/\$ $/\$ # # $\/$ #
31	14, 18, 29, 30	syl3anc 1135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
32	26, 31	orim12d 700	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$ # # $\/$ # $\/$ # $\/$ #
33	21, 32	sylbid 139	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ # $\/$ # $\/$ #
34		or4 688	. . . . . . . . . . . 12 # $\/$ # $\/$ # $\/$ # # $\/$ # $\/$ # $\/$ #
35	33, 34	syl6ib 150	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ # $\/$ # $\/$ #
36		simplr 482	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	ad3antrrr 461	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	10, 37	breq12d 3777	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
39		apreim 7594	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ # # $\/$ #
40	13, 14, 24, 29, 39	syl22anc 1136	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
41	38, 40	bitrd 177	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
42	11, 37	breq12d 3777	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
43		apreim 7594	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ # # $\/$ #
44	16, 18, 24, 29, 43	syl22anc 1136	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
45	42, 44	bitrd 177	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
46	41, 45	orbi12d 707	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$ # # $\/$ # $\/$ # $\/$ #
47	35, 46	sylibrd 158	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
48	47	ex 108	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
49	48	rexlimdvva 2440	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
50	9, 49	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
51	50	ex 108	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
52	51	rexlimdvva 2440	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
53	5, 52	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
54	53	ex 108	. . 3 $/\$ $/\$ $/\$ $/\$ # # $\/$ #
55	54	rexlimdvva 2440	. 2 $/\$ $/\$ # # $\/$ #
56	2, 55	mpd 13	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $/\$ w3a 885

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 # 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-ltwlin 6997 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

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