sotritric - Intuitionistic Logic Explorer

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Theorem sotritric 4339

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 4338	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 424	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 983	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 6	. . . . 5 $\/$
7		pm2.24 622	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 717	. . . 4 $\/$ $\/$ $\/$
9	5, 8	sylbi 121	. . 3 $\/$ $\/$ $\/$
10	4, 9	syl 14	. 2 $/\$ $\/$
11	3, 10	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $\/$ w3o 979

wceq 1364

wcel 2160 class class class wbr 4018

wor 4310

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-ext 2171

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-po 4311 df-iso 4312

This theorem is referenced by: nqtric 7416