reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8075

Description: Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)

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

**Proof of Theorem reapcotr**
Step	Hyp	Ref	Expression
1		reaplt 8065	. . . . 5 $/\$ # $\/$
2	1	3adant3 963	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 7554	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 7554	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1147	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 735	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 148	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 682	. . . . 5 $\/$ $\/$
9	8	orbi2i 714	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 724	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 182	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	syl6ib 159	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8065	. . . 4 $/\$ # $\/$
14	13	3adant2 962	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8065	. . . 4 $/\$ # $\/$
16	15	3adant1 961	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 742	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 167	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 103 $\/$ wo 664 $/\$ w3a 924

wcel 1438 class class class wbr 3845

cr 7349

clt 7522 # cap 8058

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulrcl 7444 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-precex 7455 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-apti 7460 ax-pre-ltadd 7461 ax-pre-mulgt0 7462

This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7524 df-mnf 7525 df-ltxr 7527 df-sub 7655 df-neg 7656 df-reap 8052 df-ap 8059

This theorem is referenced by: apcotr 8084