reapcotr - Intuitionistic Logic Explorer

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Theorem reapcotr 8136

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 8126	. . . . 5 $/\$ # $\/$
2	1	3adant3 964	. . . 4 $/\$ $/\$ # $\/$
3		axltwlin 7615	. . . . 5 $/\$ $/\$ $\/$
4		axltwlin 7615	. . . . . 6 $/\$ $/\$ $\/$
5	4	3com12 1148	. . . . 5 $/\$ $/\$ $\/$
6	3, 5	orim12d 736	. . . 4 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
7	2, 6	sylbid 149	. . 3 $/\$ $/\$ # $\/$ $\/$ $\/$
8		orcom 683	. . . . 5 $\/$ $\/$
9	8	orbi2i 715	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
10		or42 725	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	bitri 183	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
12	7, 11	syl6ib 160	. 2 $/\$ $/\$ # $\/$ $\/$ $\/$
13		reaplt 8126	. . . 4 $/\$ # $\/$
14	13	3adant2 963	. . 3 $/\$ $/\$ # $\/$
15		reaplt 8126	. . . 4 $/\$ # $\/$
16	15	3adant1 962	. . 3 $/\$ $/\$ # $\/$
17	14, 16	orbi12d 743	. 2 $/\$ $/\$ # $\/$ # $\/$ $\/$ $\/$
18	12, 17	sylibrd 168	1 $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104 $\/$ wo 665 $/\$ w3a 925

wcel 1439 class class class wbr 3851

cr 7410

clt 7583 # cap 8119

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-ltxr 7588 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120

This theorem is referenced by: apcotr 8145