reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8777

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 8767	. . . . 5 $/\$ # $\/$
2	1	3adant3 1043	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 8246	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 8246	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1233	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 793	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 150	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 735	. . . . 5 $\/$ $\/$
9	8	orbi2i 769	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 779	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 184	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	imbitrdi 161	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8767	. . . 4 $/\$ # $\/$
14	13	3adant2 1042	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8767	. . . 4 $/\$ # $\/$
16	15	3adant1 1041	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 800	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 169	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105 $\/$ wo 715 $/\$ w3a 1004

wcel 2202 class class class wbr 4088

cr 8030

clt 8213 # cap 8760

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pnf 8215 df-mnf 8216 df-ltxr 8218 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761

This theorem is referenced by: apcotr 8786