sotritric - Intuitionistic Logic Explorer

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Theorem sotritric 4445

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 4444	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 424	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 1008	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 6	. . . . 5 $\/$
7		pm2.24 626	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 724	. . . 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 716 $\/$ w3o 1004

wceq 1398

wcel 2203 class class class wbr 4109

wor 4416

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-ext 2214

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-v 2815 df-un 3215 df-sn 3695 df-pr 3696 df-op 3698 df-br 4110 df-po 4417 df-iso 4418

This theorem is referenced by: nqtric 7714