reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8619

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 8609	. . . . 5 $/\$ # $\/$
2	1	3adant3 1019	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 8089	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 8089	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1209	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 787	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 150	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 729	. . . . 5 $\/$ $\/$
9	8	orbi2i 763	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 773	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 184	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	imbitrdi 161	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8609	. . . 4 $/\$ # $\/$
14	13	3adant2 1018	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8609	. . . 4 $/\$ # $\/$
16	15	3adant1 1017	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 794	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 169	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105 $\/$ wo 709 $/\$ w3a 980

wcel 2164 class class class wbr 4030

cr 7873

clt 8056 # cap 8602

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603

This theorem is referenced by: apcotr 8628