Theorem sotr2 5064

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

Assertion

Ref	Expression
sotr2	|- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( -. C R B /\ C R D ) -> B R D ) )

Proof of Theorem sotr2

Step	Hyp	Ref	Expression
1		sotric 5061	. . . . . 6 |- ( ( R Or A /\ ( C e. A /\ B e. A ) ) -> ( C R B <-> -. ( C = B \/ B R C ) ) )
2	1	ancom2s 844	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( C R B <-> -. ( C = B \/ B R C ) ) )
3	2	3adantr3 1222	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( C R B <-> -. ( C = B \/ B R C ) ) )
4	3	con2bid 344	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( C = B \/ B R C ) <-> -. C R B ) )
5		breq1 4656	. . . . . 6 |- ( C = B -> ( C R D <-> B R D ) )
6	5	biimpd 219	. . . . 5 |- ( C = B -> ( C R D -> B R D ) )
7	6	a1i 11	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( C = B -> ( C R D -> B R D ) ) )
8		sotr 5057	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )
9	8	expd 452	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( B R C -> ( C R D -> B R D ) ) )
10	7, 9	jaod 395	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( C = B \/ B R C ) -> ( C R D -> B R D ) ) )
11	4, 10	sylbird 250	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( -. C R B -> ( C R D -> B R D ) ) )
12	11	impd 447	1 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( -. C R B /\ C R D ) -> B R D ) )

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

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

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-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-po 5035  df-so 5036

This theorem is referenced by:  erdszelem8  31180  nosupbnd1  31860  slelttr  31882