reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8551

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 8541	. . . . 5 $/\$ # $\/$
2	1	3adant3 1017	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 8021	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 8021	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1207	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 786	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 150	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 728	. . . . 5 $\/$ $\/$
9	8	orbi2i 762	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 772	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 184	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	syl6ib 161	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8541	. . . 4 $/\$ # $\/$
14	13	3adant2 1016	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8541	. . . 4 $/\$ # $\/$
16	15	3adant1 1015	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 793	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 169	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105 $\/$ wo 708 $/\$ w3a 978

wcel 2148 class class class wbr 4002

cr 7807

clt 7988 # cap 8534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535

This theorem is referenced by: apcotr 8560