reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8384

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 8374	. . . . 5 $/\$ # $\/$
2	1	3adant3 1002	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 7856	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 7856	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1186	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 776	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 149	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 718	. . . . 5 $\/$ $\/$
9	8	orbi2i 752	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 762	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 183	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	syl6ib 160	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8374	. . . 4 $/\$ # $\/$
14	13	3adant2 1001	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8374	. . . 4 $/\$ # $\/$
16	15	3adant1 1000	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 783	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 168	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104 $\/$ wo 698 $/\$ w3a 963

wcel 1481 class class class wbr 3937

cr 7643

clt 7824 # cap 8367

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368

This theorem is referenced by: apcotr 8393