reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8617

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 8607	. . . . 5 $/\$ # $\/$
2	1	3adant3 1019	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 8087	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 8087	. . . . . 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 8607	. . . 4 $/\$ # $\/$
14	13	3adant2 1018	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8607	. . . 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 4029

cr 7871

clt 8054 # cap 8600

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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989

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 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-ltxr 8059 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601

This theorem is referenced by: apcotr 8626