reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8360

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 8350	. . . . 5 $/\$ # $\/$
2	1	3adant3 1001	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 7832	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 7832	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1185	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 775	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 149	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 717	. . . . 5 $\/$ $\/$
9	8	orbi2i 751	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 761	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 183	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	syl6ib 160	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8350	. . . 4 $/\$ # $\/$
14	13	3adant2 1000	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8350	. . . 4 $/\$ # $\/$
16	15	3adant1 999	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 782	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 168	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104 $\/$ wo 697 $/\$ w3a 962

wcel 1480 class class class wbr 3929

cr 7619

clt 7800 # cap 8343

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 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737

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-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 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-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344

This theorem is referenced by: apcotr 8369