Theorem sotri2 5525

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
sotri2	|- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C )

Proof of Theorem sotri2

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 |- R C_ ( S X. S )
2	1	brel 5168	. . . 4 |- ( B R C -> ( B e. S /\ C e. S ) )
3	2	simpld 475	. . 3 |- ( B R C -> B e. S )
4		soi.1	. . . . . . 7 |- R Or S
5		sotric 5061	. . . . . . 7 |- ( ( R Or S /\ ( B e. S /\ A e. S ) ) -> ( B R A <-> -. ( B = A \/ A R B ) ) )
6	4, 5	mpan 706	. . . . . 6 |- ( ( B e. S /\ A e. S ) -> ( B R A <-> -. ( B = A \/ A R B ) ) )
7	6	con2bid 344	. . . . 5 |- ( ( B e. S /\ A e. S ) -> ( ( B = A \/ A R B ) <-> -. B R A ) )
8		breq1 4656	. . . . . . 7 |- ( B = A -> ( B R C <-> A R C ) )
9	8	biimpd 219	. . . . . 6 |- ( B = A -> ( B R C -> A R C ) )
10	4, 1	sotri 5523	. . . . . . 7 |- ( ( A R B /\ B R C ) -> A R C )
11	10	ex 450	. . . . . 6 |- ( A R B -> ( B R C -> A R C ) )
12	9, 11	jaoi 394	. . . . 5 |- ( ( B = A \/ A R B ) -> ( B R C -> A R C ) )
13	7, 12	syl6bir 244	. . . 4 |- ( ( B e. S /\ A e. S ) -> ( -. B R A -> ( B R C -> A R C ) ) )
14	13	com3r 87	. . 3 |- ( B R C -> ( ( B e. S /\ A e. S ) -> ( -. B R A -> A R C ) ) )
15	3, 14	mpand 711	. 2 |- ( B R C -> ( A e. S -> ( -. B R A -> A R C ) ) )
16	15	3imp231 1258	1 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   Or wor 5034   X. cxp 5112

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-ral 2917  df-rex 2918  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-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120

This theorem is referenced by:  supsrlem  9932