Theorem sotri3 5407

Description: A transitivity relation. (Read A < B and B <_ C implies A < C.) (Contributed by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	|- R Or S
soi.2	|- R C_ ( S X. S )

Assertion

Ref	Expression
sotri3	|- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Proof of Theorem sotri3

Step	Hyp	Ref	Expression
1		soi.2	. . . . . 6 |- R C_ ( S X. S )
2	1	brel 5057	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
3	2	simprd 463	. . . 4 |- ( A R B -> B e. S )
4		soi.1	. . . . . . . 8 |- R Or S
5		sotric 4835	. . . . . . . 8 |- ( ( R Or S /\ ( C e. S /\ B e. S ) ) -> ( C R B <-> -. ( C = B \/ B R C ) ) )
6	4, 5	mpan 670	. . . . . . 7 |- ( ( C e. S /\ B e. S ) -> ( C R B <-> -. ( C = B \/ B R C ) ) )
7	6	con2bid 329	. . . . . 6 |- ( ( C e. S /\ B e. S ) -> ( ( C = B \/ B R C ) <-> -. C R B ) )
8		breq2 4460	. . . . . . . 8 |- ( C = B -> ( A R C <-> A R B ) )
9	8	biimprd 223	. . . . . . 7 |- ( C = B -> ( A R B -> A R C ) )
10	4, 1	sotri 5404	. . . . . . . 8 |- ( ( A R B /\ B R C ) -> A R C )
11	10	expcom 435	. . . . . . 7 |- ( B R C -> ( A R B -> A R C ) )
12	9, 11	jaoi 379	. . . . . 6 |- ( ( C = B \/ B R C ) -> ( A R B -> A R C ) )
13	7, 12	syl6bir 229	. . . . 5 |- ( ( C e. S /\ B e. S ) -> ( -. C R B -> ( A R B -> A R C ) ) )
14	13	com3r 79	. . . 4 |- ( A R B -> ( ( C e. S /\ B e. S ) -> ( -. C R B -> A R C ) ) )
15	3, 14	mpan2d 674	. . 3 |- ( A R B -> ( C e. S -> ( -. C R B -> A R C ) ) )
16	15	com12 31	. 2 |- ( C e. S -> ( A R B -> ( -. C R B -> A R C ) ) )
17	16	3imp 1190	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   C_ wss 3471   class class class wbr 4456   Or wor 4808   X. cxp 5006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-po 4809  df-so 4810  df-xp 5014

This theorem is referenced by:  archnq  9375