Theorem sotrieq2 5063

Description: Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)

Assertion

Ref	Expression
sotrieq2	|- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> ( -. B R C /\ -. C R B ) ) )

Proof of Theorem sotrieq2

Step	Hyp	Ref	Expression
1		sotrieq 5062	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )
2		ioran 511	. 2 |- ( -. ( B R C \/ C R B ) <-> ( -. B R C /\ -. C R B ) )
3	1, 2	syl6bb 276	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   = 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:  fisupg  8208  supmo  8358  infmo  8401  fiinfg  8405  lttri3  10121  xrlttri3  11976  nosupbnd1lem2  31855  slttrieq2  31875  wessf1ornlem  39371