sotritric - Intuitionistic Logic Explorer

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Theorem sotritric 4151

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 4150	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 415	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 927	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 5	. . . . 5 $\/$
7		pm2.24 586	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 671	. . . 4 $\/$ $\/$ $\/$
9	5, 8	sylbi 119	. . 3 $\/$ $\/$ $\/$
10	4, 9	syl 14	. 2 $/\$ $\/$
11	3, 10	impbid 127	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 664 $\/$ w3o 923

wceq 1289

wcel 1438 class class class wbr 3845

wor 4122

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070

This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-po 4123 df-iso 4124

This theorem is referenced by: nqtric 6958