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Theorem sspsstri 3291
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 513 . 2  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  B  C.  A ) )
2 sspss 3288 . . . 4  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
3 sspss 3288 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
4 eqcom 2298 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
54orbi2i 505 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
63, 5bitri 240 . . . 4  |-  ( B 
C_  A  <->  ( B  C.  A  \/  A  =  B ) )
72, 6orbi12i 507 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
8 orordir 517 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
97, 8bitr4i 243 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
10 df-3or 935 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
111, 9, 103bitr4i 268 1  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    \/ w3o 933    = wceq 1632    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  ordtri3or  4440  sorpss  6298  sorpssi  6299  funpsstri  24192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
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