sotritric - Intuitionistic Logic Explorer

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Theorem sotritric 4415

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 4414	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 424	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 1005	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 6	. . . . 5 $\/$
7		pm2.24 624	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 721	. . . 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 713 $\/$ w3o 1001

wceq 1395

wcel 2200 class class class wbr 4083

wor 4386

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-v 2801 df-un 3201 df-sn 3672 df-pr 3673 df-op 3675 df-br 4084 df-po 4387 df-iso 4388

This theorem is referenced by: nqtric 7586