brtp - Mathbox for Scott Fenton

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Theorem brtp 31639

Description: A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)

**Hypotheses**
Ref	Expression
brtp.1
brtp.2

**Assertion**
Ref	Expression
brtp	$/\$ $\/$ $/\$ $\/$ $/\$

**Proof of Theorem brtp**
Step	Hyp	Ref	Expression
1		df-br 4654	. 2
2		opex 4932	. . 3
3	2	eltp 4230	. 2 $\/$ $\/$
4		brtp.1	. . . 4
5		brtp.2	. . . 4
6	4, 5	opth 4945	. . 3 $/\$
7	4, 5	opth 4945	. . 3 $/\$
8	4, 5	opth 4945	. . 3 $/\$
9	6, 7, 8	3orbi123i 1252	. 2 $\/$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	1, 3, 9	3bitri 286	1 $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wb 196 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

cvv 3200

ctp 4181

cop 4183 class class class wbr 4653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-br 4654

This theorem is referenced by: sltval2 31809 sltintdifex 31814 sltres 31815 noextendlt 31822 noextendgt 31823 nolesgn2o 31824 sltsolem1 31826 nosepnelem 31830 nosep1o 31832 nosepdmlem 31833 nodenselem8 31841 nodense 31842 nolt02o 31845 nosupbnd2lem1 31861