Theorem sotrieq 5062

Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
sotrieq	|- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Proof of Theorem sotrieq

Step	Hyp	Ref	Expression
1		sonr 5056	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
2	1	adantrr 753	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3		pm1.2 535	. . . . . 6 |- ( ( B R B \/ B R B ) -> B R B )
4	2, 3	nsyl 135	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R B \/ B R B ) )
5		breq2 4657	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
6		breq1 4656	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
7	5, 6	orbi12d 746	. . . . . 6 |- ( B = C -> ( ( B R B \/ B R B ) <-> ( B R C \/ C R B ) ) )
8	7	notbid 308	. . . . 5 |- ( B = C -> ( -. ( B R B \/ B R B ) <-> -. ( B R C \/ C R B ) ) )
9	4, 8	syl5ibcom 235	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C -> -. ( B R C \/ C R B ) ) )
10	9	con2d 129	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C \/ C R B ) -> -. B = C ) )
11		solin 5058	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C \/ B = C \/ C R B ) )
12		3orass 1040	. . . . . 6 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B R C \/ ( B = C \/ C R B ) ) )
13	11, 12	sylib 208	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C \/ ( B = C \/ C R B ) ) )
14		or12 545	. . . . 5 |- ( ( B R C \/ ( B = C \/ C R B ) ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
15	13, 14	sylib 208	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C \/ ( B R C \/ C R B ) ) )
16	15	ord 392	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( -. B = C -> ( B R C \/ C R B ) ) )
17	10, 16	impbid 202	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C \/ C R B ) <-> -. B = C ) )
18	17	con2bid 344	1 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = 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:  sotrieq2  5063  sossfld  5580  soisores  6577  soisoi  6578  weniso  6604  wemapsolem  8455  distrlem4pr  9848  addcanpr  9868  sqgt0sr  9927  lttri2  10120  xrlttri2  11975  xrltne  11994  sotrine  31658  soseq  31751