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Theorem sspsstri 3567
Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
sspsstri  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )

Proof of Theorem sspsstri
StepHypRef Expression
1 or32 529 . 2  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  B  C.  A ) )
2 sspss 3564 . . . 4  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
3 sspss 3564 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
4 eqcom 2431 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
54orbi2i 521 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
63, 5bitri 252 . . . 4  |-  ( B 
C_  A  <->  ( B  C.  A  \/  A  =  B ) )
72, 6orbi12i 523 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
8 orordir 533 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
97, 8bitr4i 255 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
10 df-3or 983 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
111, 9, 103bitr4i 280 1  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    \/ w3o 981    = wceq 1437    C_ wss 3436    C. wpss 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-ne 2616  df-in 3443  df-ss 3450  df-pss 3452
This theorem is referenced by:  ordtri3or  5474  sorpss  6590  sorpssi  6591  funpsstri  30413
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