sotritric - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > sotritric		Unicode version

Theorem sotritric 4285

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 4284	. . 3 $/\$ $/\$ $\/$
3	1, 2	mpan 421	. 2 $/\$ $\/$
4		sotritric.tri	. . 3 $/\$ $\/$ $\/$
5		3orass 966	. . . 4 $\/$ $\/$ $\/$ $\/$
6		ax-1 6	. . . . 5 $\/$
7		pm2.24 611	. . . . 5 $\/$ $\/$
8	6, 7	jaoi 706	. . . 4 $\/$ $\/$ $\/$
9	5, 8	sylbi 120	. . 3 $\/$ $\/$ $\/$
10	4, 9	syl 14	. 2 $/\$ $\/$
11	3, 10	impbid 128	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 962

wceq 1335

wcel 2128 class class class wbr 3966

wor 4256

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-un 3106 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-po 4257 df-iso 4258

This theorem is referenced by: nqtric 7320