sotritric - Intuitionistic Logic Explorer

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Theorem sotritric 4061

Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)

**Hypotheses**
Ref	Expression
sotritric.or
sotritric.tri	$/\$ $\/$ $\/$

**Assertion**
Ref	Expression
sotritric	$/\$ $\/$

**Proof of Theorem sotritric**
Step	Hyp	Ref	Expression
1		sotritric.or	. . 3
2		sotricim 4060	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 400	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 888	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 5	. . . . 5 $\/$
7		pm2.24 551	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 636	. . . 4 $\/$ $\/$ $\/$
9	5, 8	sylbi 114	. . 3 $\/$ $\/$ $\/$
10	4, 9	syl 14	. 2 $/\$ $\/$
11	3, 10	impbid 120	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $\/$ w3o 884

wceq 1243

wcel 1393 class class class wbr 3764

wor 4032

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022

This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 df-iso 4034

This theorem is referenced by: nqtric 6497